Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

**Quantum** **Mechanics**

Lecture Notes for

Chemistry 6312

**Quantum** Chemistry

**Eric** R. **Bittner**

**University** 1

**of** **Houston**

Department **of** Chemistry

Lecture Notes on **Quantum** Chemistry

Lecture notes to accompany Chemistry 6321

Copyright @1997-2003, **University** **of** **Houston** and **Eric** R. **Bittner**

All Rights Reserved.

August 12, 2003

Contents

0 Introduction 8

0.1 Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

0.2 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

0.3 2003 Course Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

I Lecture Notes 14

1 Survey **of** Classical **Mechanics** 15

1.1 Newton’s equations **of** motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.1 Elementary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.2 Phase plane analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Lagrangian **Mechanics** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 The Principle **of** Least Action . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.2 Example: 3 dimensional harmonic oscillator in spherical coordinates . . . . 20

1.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Hamiltonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 Interaction between a charged particle and an electromagnetic field. . . . . 24

1.4.2 Time dependence **of** a dynamical variable . . . . . . . . . . . . . . . . . . . 26

1.4.3 Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Waves and Wavefunctions 29

2.1 Position and Momentum Representation **of** |ψ〉 . . . . . . . . . . . . . . . . . . . . 29

2.2 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 Gaussian Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.2 Evolution **of** ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 Infinite Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.2 Particle in a finite Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Scattering states and resonances. . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.4 Application: **Quantum** Dots . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Tunneling and transmission in a 1D chain . . . . . . . . . . . . . . . . . . . . . . 49

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1

3 Semi-Classical **Quantum** **Mechanics** 55

3.1 Bohr-Sommerfield quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 The WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1 Asymptotic expansion for eigenvalue spectrum . . . . . . . . . . . . . . . . 58

3.2.2 WKB Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.3 Semi-classical Tunneling and Barrier Penetration . . . . . . . . . . . . . . 62

3.3 Connection Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.1 Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.2 Scattering at small deflection angles . . . . . . . . . . . . . . . . . . . . . . 73

3.4.3 **Quantum** treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4.4 Semiclassical evaluation **of** phase shifts . . . . . . . . . . . . . . . . . . . . 75

3.4.5 Resonance Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Postulates **of** **Quantum** **Mechanics** 80

4.0.1 The description **of** a physical state: . . . . . . . . . . . . . . . . . . . . . . 85

4.0.2 Description **of** Physical Quantities: . . . . . . . . . . . . . . . . . . . . . . 85

4.0.3 **Quantum** Measurement: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.0.4 The Principle **of** Spectral Decomposition: . . . . . . . . . . . . . . . . . . . 86

4.0.5 The Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.0.6 Reduction **of** the wavepacket: . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.0.7 The temporal evolution **of** the system: . . . . . . . . . . . . . . . . . . . . 90

4.0.8 Dirac **Quantum** Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1 Dirac Notation and Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1.1 Transformations and Representations . . . . . . . . . . . . . . . . . . . . . 94

4.1.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.3 Products **of** Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.4 Functions Involving Operators . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Constants **of** the Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Bohr Frequency and Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 Example using the particle in a box states . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Time Evolution **of** Wave and Observable . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 “Unstable States” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.7 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 Bound States **of** The Schrödinger Equation 110

5.1 Introduction to Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2 The Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.2 Constraints and Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . 114

5.2.3 Variational method applied to Schrödinger equation . . . . . . . . . . . . . 117

5.2.4 Variational theorems: Rayleigh-Ritz Technique . . . . . . . . . . . . . . . . 118

5.2.5 Variational solution **of** harmonic oscillator ground State . . . . . . . . . . . 119

5.3 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2

5.3.1 Harmonic Oscillators and Nuclear Vibrations . . . . . . . . . . . . . . . . . 124

5.3.2 Classical interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3.3 Molecular Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4 Numerical Solution **of** the Schrödinger Equation . . . . . . . . . . . . . . . . . . . 136

5.4.1 Numerov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4.2 Numerical Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6 **Quantum** **Mechanics** in 3D 152

6.1 **Quantum** Theory **of** Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.2 Eigenvalues **of** the Angular Momentum Operator . . . . . . . . . . . . . . . . . . 157

6.3 Eigenstates **of** L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4 Eigenfunctions **of** L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.5 Addition theorem and matrix elements . . . . . . . . . . . . . . . . . . . . . . . . 162

6.6 Legendre Polynomials and Associated Legendre Polynomials . . . . . . . . . . . . 164

6.7 **Quantum** rotations in a semi-classical context . . . . . . . . . . . . . . . . . . . . 165

6.8 Motion in a central potential: The Hydrogen Atom . . . . . . . . . . . . . . . . . 170

6.8.1 Radial Hydrogenic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.9 Spin 1/2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.9.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.9.2 Other Spin Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.9.3 Evolution **of** a state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.9.4 Larmor Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.10 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7 Perturbation theory 180

7.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.2 Two level systems subject to a perturbation . . . . . . . . . . . . . . . . . . . . . 182

7.2.1 Expansion **of** Energies in terms **of** the coupling . . . . . . . . . . . . . . . 183

7.2.2 Dipole molecule in homogenous electric field . . . . . . . . . . . . . . . . . 184

7.3 Dyson Expansion **of** the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . 188

7.4 Van der Waals forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.4.1 Origin **of** long-ranged attractions between atoms and molecules . . . . . . . 190

7.4.2 Attraction between an atom a conducting surface . . . . . . . . . . . . . . 192

7.5 Perturbations Acting over a Finite amount **of** Time . . . . . . . . . . . . . . . . . 193

7.5.1 General form **of** time-dependent perturbation theory . . . . . . . . . . . . 193

7.5.2 Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.6 Interaction between an atom and light . . . . . . . . . . . . . . . . . . . . . . . . 197

7.6.1 Fields and potentials **of** a light wave . . . . . . . . . . . . . . . . . . . . . 197

7.6.2 Interactions at Low Light Intensity . . . . . . . . . . . . . . . . . . . . . . 198

7.6.3 Photoionization **of** Hydrogen 1s . . . . . . . . . . . . . . . . . . . . . . . . 202

7.6.4 Spontaneous Emission **of** Light . . . . . . . . . . . . . . . . . . . . . . . . 204

7.7 Time-dependent golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.7.1 Non-radiative transitions between displaced Harmonic Wells . . . . . . . . 210

7.7.2 Semi-Classical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

3

7.8 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8 Many Body **Quantum** **Mechanics** 222

8.1 Symmetry with respect to particle Exchange . . . . . . . . . . . . . . . . . . . . . 222

8.2 Matrix Elements **of** Electronic Operators . . . . . . . . . . . . . . . . . . . . . . . 227

8.3 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

8.3.1 Two electron integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

8.3.2 Koopman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.4 **Quantum** Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.4.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . 232

8.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

A Physical Constants and Conversion Factors 247

B Mathematical Results and Techniques to Know and Love 249

B.1 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

B.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

B.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

B.1.3 Spectral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

B.2 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

B.2.1 Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

B.2.2 Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

B.2.3 Cylindrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

C Mathematica Notebook Pages 256

4

List **of** Figures

1.1 Tangent field for simple pendulum with ω = 1. The superimposed curve is a linear

approximation to the pendulum motion. . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Vector diagram for motion in a central forces. The particle’s motion is along the

Z axis which lies in the plane **of** the page. . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Screen shot **of** using Mathematica to plot phase-plane for harmonic oscillator.

Here k/m = 1 and our xo = 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1 A gaussian wavepacket, ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Momentum-space distribution **of** ψ(k). . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Go for fixed t as a function **of** x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Evolution **of** a free particle wavefunction. . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Particle in a box states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Graphical solution to transendental equations for an electron in a truncated hard

well **of** depth Vo = 10 and width a = 2. The short-dashed blue curve corresponds

to the symmetric case�and the long-dashed blue curve corresponds to the asymetric

case. The red line is

1 − V o/E. Bound state solution are such that the red and

blue curves cross. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 Transmission (blue) and Reflection (red) coefficients for an electron scattering over

a square well (V = −40 and a = 1 ). . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.8 Transmission Coefficient for particle passing over a bump. . . . . . . . . . . . . . 43

2.9 Scattering waves for particle passing over a well. . . . . . . . . . . . . . . . . . . . 44

2.10 Argand plot **of** a scattering wavefunction passing over a well. . . . . . . . . . . . . 45

2.11 Density **of** states for a 1-, 2- , and 3- dimensional space. . . . . . . . . . . . . . . . 46

2.12 Density **of** states for a quantum well and quantum wire compared to a 3d space.

Here L = 5 and s = 2 for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.13 Spherical Bessel functions, j0, j1, and j1 (red, blue, green) . . . . . . . . . . . . . 48

2.14 Radial wavefuncitons (left column) and corresponding PDFs (right column) for an

electron in a R = 0.5˚Aquantum dot. The upper two correspond to (n, l) = (1, 0)

(solid) and (n, l) = (1, 1) (dashed) while the lower correspond to (n, l) = (2, 0)

(solid) and (n, l) = (2, 1) (dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Eckart Barrier and parabolic approximation **of** the transition state . . . . . . . . . 63

3.2 Airy functions, Ai(y) (red) and Bi(y) (blue) . . . . . . . . . . . . . . . . . . . . . 66

3.3 Bound states in a graviational well . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4 Elastic scattering trajectory for classical collision . . . . . . . . . . . . . . . . . . 70

5

3.5 Form **of** the radial wave for repulsive (short dashed) and attractive (long dashed)

potentials. The form for V = 0 is the solid curve for comparison. . . . . . . . . . . 76

4.1 Gaussian distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Combination **of** two distrubitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Constructive and destructive interference from electron/two-slit experiment. The

superimposed red and blue curves are P1 and P2 from the classical probabilities . 83

4.4 The diffraction function sin(x)/x . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.1 Variational paths between endpoints. . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2 Hermite Polynomials, Hn up to n = 3. . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3 Harmonic oscillator functions for n = 0 to 3 . . . . . . . . . . . . . . . . . . . . . 132

5.4 **Quantum** and Classical Probability Distribution Functions for Harmonic Oscillator.133

5.5 London-Eyring-Polanyi-Sato (LEPS) empirical potential for the F +H2 → F H+H

chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.6 Morse well and harmonic approximation for HF . . . . . . . . . . . . . . . . . . . 136

5.7 Model potential for proton tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.8 Double well tunneling states as determined by the Numerov approach. . . . . . . . 138

5.9 Tchebyshev Polynomials for n = 1 − 5 . . . . . . . . . . . . . . . . . . . . . . . . 140

5.10 Ammonia Inversion and Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.1 Vector model for the quantum angular momentum state |jm〉, which is represented

here by the vector j which precesses about the z axis (axis **of** quantzation) with

projection m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2 Spherical Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3 Classical and **Quantum** Probability Distribution Functions for Angular Momentum.168

7.1 Variation **of** energy level splitting as a function **of** the applied field for an ammonia

molecule in an electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.2 Photo-ionization spectrum for hydrogen atom. . . . . . . . . . . . . . . . . . . . . 205

8.1 Various contributions to the H + 2 Hamiltonian. . . . . . . . . . . . . . . . . . . . . 236

8.2 Potential energy surface for H + 2 molecular ion. . . . . . . . . . . . . . . . . . . . . 238

8.3 Three dimensional representations **of** ψ+ and ψ− for the H + 2 molecular ion. . . . . 238

8.4 Setup calculation dialog screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.5 HOMO-1, HOMO and LUMO for CH2 = O. . . . . . . . . . . . . . . . . . . . . . 245

8.6 Transition state geometry for H2 + C = O → CH2 = O. The Arrow indicates the

reaction path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

B.1 sin(xa)/πx representation **of** the Dirac δ-function . . . . . . . . . . . . . . . . . . 251

B.2 Gaussian representation **of** δ-function . . . . . . . . . . . . . . . . . . . . . . . . . 251

6

List **of** Tables

3.1 Location **of** nodes for Airy, Ai(x) function. . . . . . . . . . . . . . . . . . . . . . . 68

5.1 Tchebychev polynomials **of** the first type . . . . . . . . . . . . . . . . . . . . . . . 140

5.2 Eigenvalues for double well potential computed via DVR and Numerov approaches 143

6.1 Spherical Harmonics (Condon-Shortley Phase convention. . . . . . . . . . . . . . . 160

6.2 Relation between various notations for Clebsch-Gordan Coefficients in the literature169

8.1 Vibrational Frequencies **of** Formaldehyde . . . . . . . . . . . . . . . . . . . . . . . 244

A.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

A.2 Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

A.3 Useful orders **of** magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

7

Chapter 0

Introduction

Nothing conveys the impression **of** humungous intellect so much as even the sketchiest

knowledge **of** quantum physics, and since the sketchiest knowledge is all anyone will

ever have, never be shy about holding forth with bags **of** authority about subatomic

particles and the quantum realm without having done any science whatsoever.

Jack Klaff –Bluff Your Way in the **Quantum** Universe

The field **of** quantum chemistry seeks to provide a rigorous description **of** chemical processes

at its most fundamental level. For ordinary chemical processes, the most fundamental and underlying

theory **of** chemistry is given by the time-dependent and time-independent version **of** the

Schrödinger equation. However, simply stating an equation that provides the underlying theory

in now shape or form yields and predictive or interpretive power. In fact, most **of** what we do

in quantum mechanics is to develop a series **of** well posed approximation and physical assumptions

to solve basic equations **of** quantum mechanics. In this course, we will delve deeply into

the underlying physical and mathematical theory. We will learn how to solve some elementary

problems and apply these to not so elementary examples.

As with any course **of** this nature, the content reflects the instructors personal interests in

the field. In this case, the emphasis **of** the course is towards dynamical processes, transitions

between states, and interaction between matter and radiation. More “traditional” quantum

chemistry courses will focus upon electronic structure. In fact, the moniker “quantum chemistry”

typically refers to electronic structure theory. While this is an extremely rich topic, it is my

personal opinion that a deeper understanding **of** dynamical processes provides a broader basis

for understanding chemical processes.

It is assumed from the beginning, that students taking this course have had some exposure

to the fundamental principles **of** quantum theory as applied to chemical systems. This is usually

in the context **of** a physical chemistry course or a separate course in quantum chemistry. I

also assume that students taking this course have had undergraduate level courses in calculus,

differential equations, and have some concepts **of** linear algebra. Students lacking in any **of** these

areas are strongly encouraged to sit through my undergraduate Physical Chemistry II course

(**of**fered in the Spring Semester at the Univ. **of** **Houston**) before attempting this course. This

course is by design and purpose theoretical in nature.

The purpose **of** this course is to provide a solid and mathematically rigorous tour through

modern quantum mechanics. We will begin with simple examples which can be worked out

8

exactly on paper and move on to discuss various approximation schemes. For cases in which

analytical solutions are either too obfuscating or impossible, computer methods will be introduced

using Mathematica. Applications toward chemically relevant topics will be emphasized

throughout.

We will primarily focus upon single particle systems, or systems in which the particles are

distinguishable. Special considerations for systems **of** indistinguishable particles, such as the

electrons in a molecule, will be discussed towards the end **of** the course. The pace **of** the course

is fairly rigorous, with emphasis on solving problems either analytically or using computer.

I also tend to emphasize how to approach a problem from a theoretical viewpoint. As you

will discover rapidly, very few **of** the problem sets in this course are **of** the “look-up the right

formula” type. Rather, you will need to learn to use the various techniques (perturbation theory,

commutation relations, etc...) to solve and work out problems for a variety **of** physical systems.

The lecture notes in this package are really to be regarded as a work in progress and updates

and additions will be posted as they evolve. Lacking is a complete chapter on the Hydrogen

atom and atomic physics and a good overview **of** many body theory. Also, I have not included

a chapter on scattering and other topics as these will be added over the course **of** time. Certain

sections are clearly better than others and will be improved upon over time. Each chapter ends

with a series **of** exercises and suggested problems Some **of** which have detailed solutions. Others,

you should work out on your own. At the end **of** this book are a series **of** Mathematica notebooks

I have written which illustrate various points and perform a variety **of** calculations. These can be

downloaded from my web-site (http://k2.chem.uh.edu/quantum/) and run on any recent version

**of** Mathematica. (≥ v4.n).

It goes entirely without saying (but I will anyway) that these notes come from a wide variety

**of** sources which I have tried to cite where possible.

9

0.1 Essentials

• Instructor: **Pr of**.

• Office: Fleming 221 J

• Email: bittner@uh.edu

• Phone: -3-2775

• Office Hours: Monday and Thurs. afternoons or by appointment.

• Course Web Page: http://k2.chem.uh.edu/quantum/ Solution sets, course news, class

notes, sample computer routines, etc...will be posted from time to time on this web-page.

• Other Required Text: **Quantum** **Mechanics**, Landau and Lifshitz. This is volume 3 **of**

L&L’s classical course in modern physics. Every self-respecting scientist has at least two or

three **of** their books on their book-shelves. This text tends to be pretty terse and uses the

classical phrase it is easy to show... quite a bit (it usually means just the opposite). The

problems are usually worked out in detail and are usually classic applications **of** quantum

theory. This is a land-mark book and contains everything you really need to know to do

quantum mechanics.

• Recommended Texts: I highly recommend that you use a variety **of** books since one

author’s approach to a given topic may be clearer than another’s approach.

– **Quantum** **Mechanics**, Cohen-Tannoudji, et al. This two volume book is very comprehensive

and tends to be rather formal (and formidable) in its approach. The problems

are excellent.

–

– Lectures in **Quantum** **Mechanics**, Gordon Baym. Baym’s book covers a wide range **of**

topics in a lecture note style.

– **Quantum** Chemistry, I. Levine. This is usually the first quantum book that chemists

get. I find it to be too wordy and the notation and derivations a bit ponderous. Levine

does not use Dirac notation. However, he does give a good overview **of** elementary

electronic structure theory and some if its important developments. Good for starting

**of**f in electronic structure.

– Modern **Quantum** **Mechanics**, J. J. Sakurai. This is a real classic. Not good for a first

exposure since it assumes a fairly sophisticated understanding **of** quantum mechanics

and mathematics.

– Intermediate **Quantum** **Mechanics**, Hans Bethe and Roman Jackiw. This book is a

great exploration **of** advanced topics in quantum mechanics as illustrated by atomic

systems.

10

– What is **Quantum** **Mechanics**?, Transnational College **of** LEX. OK, this one I found

at Barnes and Noble and it’s more or less a cartoon book. But, it is really good. It

explores the historical development **of** quantum mechanics, has some really interesting

insights into semi-classical and ”old” quantum theory, and presents the study **of**

quantum mechanics as a unfolding story. This book I highly recommend if this

is the first time you are taking a course on quantum mechanics.

– **Quantum** **Mechanics** in Chemistry by George Schatz and Mark Ratner. Ratner and

Schatz have more in terms **of** elementary quantum chemistry, emphasizing the use **of**

modern quantum chemical computer programs than almost any text I have reviewed.

• Prequisites: Graduate status in chemistry. This course is required for all Physical Chemistry

graduate students. The level **of** the course will be fairly rigorous and I assume that students

have had some exposure to quantum mechanics at the undergraduate level–typically

in Physical Chemistry, and are competent in linear algebra, calculus, and solving elementary

differential equations.

• Tests and Grades: There are no exams in this course, only problem sets and participation

in discussion. This means coming to lecture prepared to ask and answer questions. My

grading policy is pretty simple. If you make an honest effort, do the assigned problems

(mostly correctly), and participate in class, you will be rewarded with at least a B. Of

course this is the formula for success for any course.

0.2 Problem Sets

Your course grade will largely be determined by your performance on these problems as well as

the assigned discussion **of** a particular problem. My philosophy towards problem sets is that this

is the only way to really learn this material. These problems are intentionally challenging, but

not overwhelming, and are paced to correspond to what will be going on in the lecture.

Some ground rules:

1. Due dates are posted on each problem–usually 1 week or 2 weeks after they are assigned.

Late submissions may be turned in up to 1 week later. All problems must be turned in by

December 3. I will not accept any submissions after that date.

2. Handwritten Problems. If I can’t read it, I won’t grade it. Period. Consequently, I strongly

encourage the use **of** word processing s**of**tware for your final submission. Problem solutions

can be submitted electronically as Mathematica, Latex, or PDF files to bittner@uh.edu with

the subject: QUANTUM PROBLEM SET. Do not send me a MSWord file as an email

attachment. I expect some text (written in compete and correct sentences) to explain your

steps where needed and some discussion **of** the results. The computer lab in the basement

**of** Fleming has 20 PCs with copies **of** Mathematica or you can obtain your own license

from the **University** Media Center.

3. Collaborations. You are strongly encouraged to work together and collaborate on problems.

However, simply copying from your fellow student is not an acceptable collaboration.

11

4. These are the only problems you need to turn in. We will have additional exercises–mostly

coming from the lecture. Also, at the end **of** the lectures herein, are a set **of** suggested

problems and exercises to work on. Many **of** these have solutions.

12

0.3 2003 Course Calendar

This is a rough schedule **of** topics we will cover. In essence we will follow the starting from

a basic description **of** quantum wave mechanics and bound states. We will then move onto

the more formal aspects **of** quantum theory: Dirac notation, perturbation theory, variational

theory, and the like. Lastly, we move onto applications: Hydrogen atom, many-electron systems,

semi-classical approximations, and a semi-classical treatment **of** light absorption and emission.

We will also have a recitation session in 221 at 10am Friday morning. The purpose **of** this

will be to specifically discuss the problem sets and other issues.

• 27-August: Course overview: Classical Concepts

• 3-Sept: Finishing Classical **Mechanics**/Elementary **Quantum** concepts

• 8-Sept: Particle in a box and hard wall potentials (Perry?)

• 10 Sept: Tunneling/Density **of** states (Perry)

• 15/17 Bohr-Sommerfield Quantization/Old quantum theory/connection to classical mechanics

(Perry)

• 22/24 Sept: Semiclassical quantum mechanics: WKB Approx. Application to scattering

• 29 Sept/1 Oct. Postulates **of** quantum mechanics: Dirac notation, superposition principle,

simple calculations.

• 6/8 Oct: Bound States: Variational principle, quantum harmonic oscillator

• 13/15 Oct: **Quantum** mechanics in 3D: Angular momentum (Chapt 4.1-4.8)

• 20/22 Oct: Hydrogen atom/Hydrogenic systems/Atomic structure

• 27/29 Oct: Perturbation Theory:

• 3/5 Nov: Time-dependent Perturbation Theory:

• 10/12 Identical Particles/**Quantum** Statistics

• 17/19 Nov: Helium atom, hydrogen ion

• 24/26 Nov: **Quantum** Chemistry

• 3 Dec–Last day to turn in problem sets

• Final Exam: TBA

13

Part I

Lecture Notes

14

Chapter 1

Survey **of** Classical **Mechanics**

**Quantum** mechanics is in many ways the cumulation **of** many hundreds **of** years **of** work and

thought about how mechanical things move and behave. Since ancient times, scientists have

wondered about the structure **of** matter and have tried to develop a generalized and underlying

theory which governs how matter moves at all length scales.

For ordinary objects, the rules **of** motion are very simple. By ordinary, I mean objects that

are more or less on the same length and mass scale as you and I, say (conservatively) 10 −7 m to

10 6 m and 10 −25 g to 10 8 g moving less than 20% **of** the speed **of** light. On other words, almost

everything you can see and touch and hold obey what are called “classical” laws **of** motion. The

term “classical” means that that the basic principles **of** this class **of** motion have their foundation

in antiquity. Classical mechanics is a extremely well developed area **of** physics. While you may

think that given that classical mechanics has been studied extensively for hundreds **of** years

there really is little new development in this field, it remains a vital and extremely active area **of**

research. Why? Because the majority **of** universe “lives” in a dimensional realm where classical

mechanics is extremely valid. Classical mechanics is the workhorse for atomistic simulations

**of** fluids, proteins, polymers. It provides the basis for understanding chaotic systems. It also

provides a useful foundation **of** many **of** the concepts in quantum mechanics.

**Quantum** mechanics provides a description **of** how matter behaves at very small length and

mass scales: i.e. the realm **of** atoms, molecules, and below. It was developed over the last century

to explain a series **of** experiments on atomic systems that could not be explained using purely

classical treatments. The advent **of** quantum mechanics forced us to look beyond the classical

theories. However, it was not a drastic and complete departure. At some point, the two theories

must correspond so that classical mechanics is the limiting behavior **of** quantum mechanics for

macroscopic objects. Consequently, many **of** the concepts we will study in quantum mechanics

have direct analogs to classical mechanics: momentum, angular momentum, time, potential

energy, kinetic energy, and action.

Much like classical music is in a particular style, classical mechanics is based upon the principle

that the motion **of** a body can be reduced to the motion **of** a point particle with a given mass

m, position x, and velocity v. In this chapter, we will review some **of** the concepts **of** classical

mechanics which are necessary for studying quantum mechanics. We will cast these in form

whereby we can move easily back and forth between classical and quantum mechanics. We will

first discuss Newtonian motion and cast this into the Lagrangian form. We will then discuss the

principle **of** least action and Hamiltonian dynamics and the concept **of** phase space.

15

1.1 Newton’s equations **of** motion

Newton’s Principia set the theoretical basis **of** mathematical mechanics and analysis **of** physical

bodies. The equation that force equals mass times acceleration is the fundamental equation **of**

classical mechanics. Stated mathematically

m¨x = f(x) (1.1)

The dots refer to differentiation with respect to time. We will use this notion for time derivatives.

We may also use x ′ or dx/dt as well. So,

¨x = d2x .

dt2 For now we are limiting our selves to one particle moving in one dimension. For motion in

more dimensions, we need to introduce vector components. In cartesian coordinates, Newton’s

equations are

m¨x = fx(x, y, z) (1.2)

m¨y = fy(x, y, z) (1.3)

m¨z = fz(x, y, z) (1.4)

where the force vector � f(x, y, z) has components in all three dimensions and varies with location.

We can also define a position vector, �x = (x, y, z), and velocity vector �v = ( ˙x, ˙y, ˙z). We can also

replace the second-order differential equation with two first order equations.

˙x = vx (1.5)

˙vx = fx/m (1.6)

These, along with the initial conditions, x(0) and v(0) are all that are needed to solve for the

motion **of** a particle with mass m given a force f. We could have chosen two end points as well

and asked, what path must the particle take to get from one point to the next. Let us consider

some elementary solutions.

1.1.1 Elementary solutions

First the case in which f = 0 and ¨x = 0. Thus, v = ˙x = const. So, unless there is an applied

force, the velocity **of** a particle will remain unchanged.

Second, we consider the case **of** a linear force, f = −kx. This is restoring force for a spring

and such force laws are termed Hooke’s law and k is termed the force constant. Our equations

are:

˙x = vx (1.7)

˙vx = −k/mx (1.8)

or ¨x = −(k/m)x. So we want some function which is its own second derivative multiplied by

some number. The cosine and sine functions have this property, so let’s try

x(t) = A cos(at) + B sin(bt).

16

Taking time derivatives

˙x(t) = −aA sin(at) + bB cos(bt);

¨x(t) = −a 2 A cos(at) − b 2 B sin(bt).

�

So we get the required result if a = b = k/m, leaving A and B undetermined. Thus, we need

two initial conditions to specify these coefficients. Let’s pick x(0) = xo �

and v(0) = 0. Thus,

x(0) = A = xo and B = 0. Notice that the term k/m has units **of** angular frequency.

So, our equation **of** motion are

v

3

2

1

0

-1

-2

-3

�

k

ω =

m

x(t) = xo cos(ωt) (1.9)

v(t) = −xoω sin(ωt). (1.10)

-2 p -p 0 p 2 p

-2 p -p 0 p 2 p

x

Figure 1.1: Tangent field for simple pendulum with ω = 1. The superimposed curve is a linear

approximation to the pendulum motion.

1.1.2 Phase plane analysis

Often one can not determine the closed form solution to a given problem and we need to turn

to more approximate methods or even graphical methods. Here, we will look at an extremely

useful way to analyze a system **of** equations by plotting their time-derivatives.

First, let’s look at the oscillator we just studied. We can define a vector s = ( ˙x, ˙v) =

(v, −k/mx) and plot the vector field. Fig. 1.3 shows how to do this in Mathematica. The

17

superimposed curve is one trajectory and the arrows give the “flow” **of** trajectories on the phase

plane.

We can examine more complex behavior using this procedure. For example, the simple

pendulum obeys the equation ¨x = −ω 2 sin x. This can be reduced to two first order equations:

˙x = v and ˙v = −ω 2 sin(x).

We can approximate the motion **of** the pendulum for small displacements by expanding the

pendulum’s force about x = 0,

−ω 2 sin(x) = −ω 2 (x − x3

6

For small x the cubic term is very small, and we have

˙v = −ω 2 x = − k

m x

+ · · ·).

which is the equation for harmonic motion. So, for small initial displacements, we see that the

pendulum oscillates back and forth with an angular frequency ω. For large initial displacements,

xo = π or if we impart some initial velocity on the system vo > 1, the pendulum does not oscillate

back and forth, but undergoes librational motion (spinning!) in one direction or the other.

1.2 Lagrangian **Mechanics**

1.2.1 The Principle **of** Least Action

The most general form **of** the law governing the motion **of** a mass is the principle **of** least action

or Hamilton’s principle. The basic idea is that every mechanical system is described by a single

function **of** coordinate, velocity, and time: L(x, ˙x, t) and that the motion **of** the particle is such

that certain conditions are satisfied. That condition is that the time integral **of** this function

S =

� tf

to

L(x, ˙x, t)dt

takes the least possible value give a path that starts at xo at the initial time and ends at xf at

the final time.

Lets take x(t) be function for which S is minimized. This means that S must increase for

any variation about this path, x(t) + δx(t). Since the end points are specified, δx(0) = δx(t) = 0

and the change in S upon replacement **of** x(t) with x(t) + δx(t) is

δS =

� tf

to

L(x + δx, ˙x + δ ˙x, t)dt −

� tf

to

L(x, ˙x, t)dt = 0

This is zero, because S is a minimum. Now, we can expand the integrand in the first term

� �

∂L ∂L

L(x + δx, ˙x + δ ˙x, t) = L(x, ˙x, t) + δx + δ ˙x

∂x ∂ ˙x

Thus, we have

� tf

to

� �

∂L ∂L

δx + δ ˙x dt = 0.

∂x ∂ ˙x

18

Since δ ˙x = dδx/dt and integrating the second term by parts

δS =

�

∂L

δ ˙x δx

�tf to

+

� tf

to

� ∂L

∂x

�

d ∂L

− δxdt = 0

dt ∂ ˙x

The surface term vanishes because **of** the condition imposed above. This leaves the integral. It

too must vanish and the only way for this to happen is if the integrand itself vanishes. Thus we

have the

∂L d ∂L

− = 0

∂x dt ∂ ˙x

L is known as the Lagrangian. Before moving on, we consider the case **of** a free particle.

The Lagrangian in this case must be independent **of** the position **of** the particle since a freely

moving particle defines an inertial frame. Since space is isotropic, L must only depend upon the

magnitude **of** v and not its direction. Hence,

L = L(v 2 ).

Since L is independent **of** x, ∂L/∂x = 0, so the Lagrange equation is

d ∂L

dt ∂v

= 0.

So, ∂L/∂v = const which leads us to conclude that L is quadratic in v. In fact,

which is the kinetic energy for a particle.

L = 1

m v2 ,

T = 1

2 mv2 = 1

2 m ˙x2 .

For a particle moving in a potential field, V , the Lagrangian is given by

L = T − V.

L has units **of** energy and gives the difference between the energy **of** motion and the energy **of**

location.

This leads to the equations **of** motion:

d ∂L

dt ∂v

= ∂L

∂x .

Substituting L = T − V , yields

m ˙v = − ∂V

∂x

which is identical to Newton’s equations given above once we identify the force as the minus the

derivative **of** the potential. For the free particle, v = const. Thus,

S =

� tf

to

m

2 v2 dt = m

2 v2 (tf − to).

19

You may be wondering at this point why we needed a new function and derived all this from

some minimization principle. The reason is that for some systems we have constraints on the type

**of** motion they can undertake. For example, there may be bonds, hinges, and other mechanical

hinderances which limit the range **of** motion a given particle can take. The Lagrangian formalism

provides a mechanism for incorporating these extra effects in a consistent and correct way. In

fact we will use this principle later in deriving a variational solution to the Schrodinger equation

by constraining the wavefunction solutions to be orthonormal.

Lastly, it is interesting to note that v 2 = (dl/d) 2 = (dl) 2 /(dt) 2 is the square **of** the element

**of** an arc in a given coordinate system. Thus, within the Lagrangian formalism it is easy to

convert from one coordinate system to another. For example, in cartesian coordinates: dl 2 =

dx 2 + dy 2 + dz 2 . Thus, v 2 = ˙x 2 + ˙y 2 + ˙z 2 . In cylindrical coordinates, dl = dr 2 + r 2 dφ 2 + dz 2 , we

have the Lagrangian

L = 1

2 m( ˙r2 + r 2 ˙ φ 2 + ˙z 2 )

and for spherical coordinates dl 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 ; hence

L = 1

2 m( ˙r2 + r 2 ˙ θ 2 + r 2 sin 2 θ ˙ φ 2 ).

1.2.2 Example: 3 dimensional harmonic oscillator in spherical coordinates

Here we take the potential energy to be a function **of** r alone (isotropic)

V (r) = kr 2 /2.

Thus, the Lagrangian in cartesian coordinates is

L = m

2 ( ˙x2 + ˙y 2 + ˙z 2 ) + k

2 r2

since r 2 = x 2 + y 2 + z 2 , we could easily solve this problem in cartesian space since

L = m

=

2 ( ˙x2 + ˙y 2 + ˙z 2 ) + k

2 (x2 + y 2 + z 2 ) (1.11)

�

�

�

� m

2 ˙x2 + k

2 x2

+

� m

2 ˙y2 + k

2 y2

+

� m

2 ˙z2 + k

2 z2

(1.12)

and we see that the system is separable into 3 independent oscillators. To convert to spherical

polar coordinates, we use

and the arc length given above.

x = r sin(φ) cos(θ) (1.13)

y = r sin(φ) sin(θ) (1.14)

z = r cos(θ) (1.15)

L = m

2 ( ˙r2 + r 2 ˙ θ 2 + r 2 sin 2 θ ˙ φ 2 ) − k

2 r2

20

The equations **of** motion are

d ∂L

dt ∂ ˙ ∂L

−

φ ∂φ

d ∂L

dt ∂ ˙ ∂L

−

θ ∂θ

d ∂L ∂L

−

dt ∂ ˙r ∂r

= d

dt (mr2 sin 2 θ ˙ φ = 0 (1.16)

= d

dt (mr2 ˙ θ) − mr 2 sin θ cos θ ˙ φ = 0 (1.17)

= d

dt (m ˙r) − mr ˙ θ 2 − mr sin 2 θ ˙ φ 2 + kr = 0 (1.18)

We now prove that the motion **of** a particle in a central force field lies in a plane containing

the origin. The force acting on the particle at any given time is in a direction towards the origin.

Now, place an arbitrary cartesian frame centered about the particle with the z axis parallel to

the direction **of** motion as sketched in Fig. 1.2 Note that the y axis is perpendicular to the plane

**of** the page and hence there is no force component in that direction. Consequently, the motion

**of** the particle is constrained to lie in the zx plane, i.e. the plane **of** the page and there is no

force component which will take the particle out **of** this plane.

Let’s make a change **of** coordinates by rotating the original frame to a new one whereby the

new z ′ is perpendicular to the plane containing the initial position and velocity vectors. In the

sketch above, this new z ′ axis would be perpendicular to the page and would contain the y axis

we placed on the moving particle. In terms **of** these new coordinates, the Lagrangian will have

the same form as before since our initial choice **of** axis was arbitrary. However, now, we have

some additional constraints. Because the motion is now constrained to lie in the x ′ y ′ plane,

θ ′ = π/2 is a constant, and ˙ θ = 0. Thus cos(π/2) = 0 and sin(π/2) = 1 in the equations above.

From the equations for φ we find

d

dt mr2 ˙ φ = 0

or

mr 2 ˙ φ = const = pφ.

This we can put into the r equation

o

d

dt (m ˙r) − mr ˙ φ 2 + kr = 0 (1.19)

d

dt (m ˙r) − p2φ + kr

mr3 = 0 (1.20)

Figure 1.2: Vector diagram for motion in a central forces. The particle’s motion is along the Z

axis which lies in the plane **of** the page.

a

21

F

Z

Y

X

Z'

Y

Z

Y'

where we notice that −p 2 φ/mr 3 is the centrifugal force. Taking the last equation, multiplying by

˙r and then integrating with respect to time gives

i.e.

X'

˙r 2 = − p2 φ

m 2 r 2 − kr2 + b (1.21)

˙r =

Integrating once again with respect to time,

�

t − to =

=

X

− p2 φ

m 2 r 2 − kr2 + b (1.22)

� rdr

�

= 1

�

2

˙r

�

rdr

− p2 φ

m 2 − kr 4 + br 2

dx

√ a + bx + cx 2

(1.23)

(1.24)

(1.25)

where x = r 2 , a = −p 2 φ/m 2 , b is the constant **of** integration, and c = −k This is a standard

integral and we can evaluate it to find

where

r 2 = 1

2ω (b + A sin(ω(t − to))) (1.26)

�

A =

m2 .

What we see then is that r follows an elliptical path in a plane determined by the initial velocity.

b 2 − ω2 p 2 φ

22

This example also illustrates another important point which has tremendous impact on molecular

quantum mechanics, namely, the angular momentum about the axis **of** rotation is conserved.

We can choose any axis we want. In order to avoid confusion, let us define χ as the angular

rotation about the body-fixed Z ′ axis and φ as angular rotation about the original Z axis. So

our conservation equations are

mr 2 ˙χ = pχ

about the Z ′ axis and

mr 2 sin θ ˙ φ = pφ

for some arbitrary fixed Z axis. The angle θ will also have an angular momentum associated

with it pθ = mr 2 ˙ θ, but we do not have an associated conservation principle for this term since it

varies with φ. We can connect pχ with pθ and pφ about the other axis via

Consequently,

pχdχ = pθdθ + pφdφ.

mr 2 ˙χ 2 dχ = mr 2 ( ˙ φ sin θdφ + ˙ θdθ).

Here we see that the the angular momentum vector remains fixed in space in the absence **of**

any external forces. Once an object starts spinning, its axis **of** rotation remains pointing in a

given direction unless something acts upon it (torque), in essence in classical mechanics we can

fully specify Lx, Ly, and Lz as constants **of** the motion since d � L/dt = 0. In a later chapter, we

will cover the quantum mechanics **of** rotations in much more detail. In the quantum case, we

will find that one cannot make such a precise specification **of** the angular momentum vector for

systems with low angular momentum. We will, however, recover the classical limit in end as we

consider the limit **of** large angular momenta.

1.3 Conservation Laws

We just encountered one extremely important concept in mechanics, namely, that some quantities

are conserved if there is an underlying symmetry. Next, we consider a conservation law arising

from the homogeneity **of** time. For a closed dynamical system, the Lagrangian does not explicitly

depend upon time. Thus we can write

dL

dt

= ∂L

∂x

∂L

˙x + ¨x (1.27)

∂ ˙x

Replacing ∂L/∂x with Lagrange’s equation, we obtain

dL

dt

=

� �

d ∂L

˙x +

dt ∂ ˙x

∂L

=

¨x

∂ ˙x

(1.28)

d

�

˙x

dt

∂L

�

∂ ˙x

(1.29)

Now, rearranging this a bit,

�

d

˙x

dt

∂L

�

− L = 0. (1.30)

∂ ˙x

23

So, we can take the quantity in the parenthesis to be a constant.

�

E = ˙x ∂L

�

− L = const. (1.31)

∂ ˙x

is an integral **of** the motion. This is the energy **of** the system. Since L can be written in form

L = T − V where T is a quadratic function **of** the velocities, and using Euler’s theorem on

homogeneous functions:

This gives,

˙x ∂L

∂ ˙x

= ˙x∂T

∂ ˙x

E = T + V

= 2T.

which says that the energy **of** the system can be written as the sum **of** two different terms: the

kinetic energy or energy **of** motion and the potential energy or the energy **of** location.

One can also prove that linear momentum is conserved when space is homogeneous. That is,

when we can translate our system some arbitrary amount ɛ and our dynamical quantities must

remain unchanged. We will prove this in the problem sets.

1.4 Hamiltonian Dynamics

Hamiltonian dynamics is a further generalization **of** classical dynamics and provides a crucial link

with quantum mechanics. Hamilton’s function, H, is written in terms **of** the particle’s position

and momentum, H = H(p, q). It is related to the Lagrangian via

Taking the derivative **of** H w.r.t. x

H = ˙xp − L(x, ˙x)

∂H

∂x

= −∂L

∂x

= − ˙p

Differentiation with respect to p gives

∂H

= ˙q.

∂p

These last two equations give the conservation conditions in the Hamiltonian formalism. If H

is independent **of** the position **of** the particle, then the generalized momentum, p is constant in

time. If the potential energy is independent **of** time, the Hamiltonian gives the total energy **of**

the system,

H = T + V.

1.4.1 Interaction between a charged particle and an electromagnetic

field.

We consider here a free particle with mass m and charge e in an electromagnetic field. The

Hamiltonian is

H = px ˙x + py ˙y + pz ˙z − L (1.32)

= ˙x ∂L

∂ ˙x

+ ˙y ∂L

∂ ˙y

24

+ ˙z ∂L

∂ ˙z

− L. (1.33)

Our goal is to write this Hamiltonian in terms **of** momenta and coordinates.

For a charged particle in a field, the force acting on the particle is the Lorenz force. Here it

is useful to introduce a vector and scaler potential and to work in cgs units.

�F = e

c �v × (� ∇ × � A) − e ∂

c

� A

∂t − e� ∇φ.

The force in the x direction is given by

Fx = d

�

e

m ˙x = ˙y

dt c

∂Ay

�

∂Az

+ ˙z −

∂x ∂x

e

�

˙y

c

∂Ax

�

∂Ax ∂Ax

+ ˙z + − e

∂y ∂z ∂t

∂φ

∂x

with the remaining components given by cyclic permutation. Since

dAx ∂Ax

∂Ax ∂Ax

= + ˙x∂Ax + ˙y + ˙z

dt ∂t ∂x ∂y ∂z ,

Fx = e

�

+ ˙x

c

∂Ax

�

∂Ax ∂Ax

+ ˙y + ˙z −

∂x ∂y ∂z

e

c �v · � A − eφ.

Based upon this, we find that the Lagrangian is

L = 1 1 1

m ˙x2 m ˙y2

2 2 2 m ˙z2 + e

c �v · � A − eφ

where φ is a velocity independent and static potential.

Continuing on, the Hamiltonian is

H = m

2 ( ˙x2 + ˙y 2 + ˙z 2 ) + eφ (1.34)

= 1

2m ((m ˙x)2 + (m ˙y) 2 + (m ˙y) 2 ) + eφ (1.35)

The velocities, m ˙x, are derived from the Lagrangian via the canonical relation

From this we find,

and the resulting Hamiltonian is

H = 1

�� px −

2m

e

c Ax

p = ∂L

∂ ˙x

m ˙x = px − e

c Ax

m ˙y = py − e

c Ay

m ˙z = pz − e

c Az

� 2

�

+ py − e

c Ay

� 2

�

+ pz − e

c Az

� �

2

+ eφ.

(1.36)

(1.37)

(1.38)

We see here an important concept relating the velocity and the momentum. In the absence **of** a

vector potential, the velocity and the momentum are parallel. However, when a vector potential

is included, the actual velocity **of** a particle is no longer parallel to its momentum and is in fact

deflected by the vector potential.

25

1.4.2 Time dependence **of** a dynamical variable

On **of** the important applications **of** Hamiltonian mechanics is in the dynamical evolution **of** a

variable which depends upon p and q, G(p, q). The total derivative **of** G is

dG

dt

= ∂G

∂t

+ ∂G

∂q

˙q + ∂G

∂p ˙p

From Hamilton’s equations, we have the canonical definitions

Thus,

dG

dt

dG

dt

˙q = ∂H

, ˙p = −∂H

∂p ∂q

∂G

=

∂t

∂G

=

∂t

∂G ∂H

+

∂q ∂p

∂G ∂H

−

∂p ∂q

(1.39)

+ {G, H}, (1.40)

where {A, B} is called the Poisson bracket **of** two dynamical quantities, G and H.

{G, H}, = ∂G ∂H

∂q ∂p

∂G ∂H

−

∂p ∂q

We can also define a linear operator L as generating the Poisson bracket with the Hamiltonian:

LG = 1

{H, G}

i

so that if G does not depend explicitly upon time,

G(t) = exp(iLt)G(0).

where exp(iLt) is the propagator which carried G(0) to G(t).

Also, note that if {G, H} = 0, then dG/dt = 0 so that G is a constant **of** the motion. This

too, along with the construction **of** the Poisson bracket has considerable importance in the realm

**of** quantum mechanics.

1.4.3 Virial Theorem

Finally, we turn our attention to a concept which has played an important role in both quantum

and classical mechanics. Consider a function G that is a product **of** linear momenta and

coordinate,

G = pq.

The time derivative is simply.

G

dt

= q ˙p + p ˙q

26

Now, let’s take a time average **of** both sides **of** this last equation.

�

d

dt pq

�

1

= lim

T →∞

= lim

T →∞

= lim

T →∞

� T

�

d

dt pq

�

dt (1.41)

T 0

�

1 T

d(pq) (1.42)

T 0

1

T ((pq)T − (pq)0) (1.43)

If the trajectories **of** system are bounded, both p and q are periodic in time and are therefore

finite. Thus, the average must vanish as T → ∞ giving

Since p ˙q = 2T and ˙p = −F , we have

〈p ˙q + q ˙p〉 = 0 (1.44)

〈2T 〉 = −〈qF 〉. (1.45)

In cartesian coordinates this leads to

� �

�

〈2T 〉 = − xiFi . (1.46)

i

For a conservative system F = −∇V . Thus, if we have a centro-symmetric potential given

by V = Cr n , it is easy to show that

〈2T 〉 = n〈V 〉.

For the case **of** the Harmonic oscillator, n = 2 and 〈T 〉 = 〈V 〉. So, for example, if we have a

total energy equal to kT in this mode, then 〈T 〉 + 〈V 〉 = kT and 〈T 〉 = 〈V 〉 = kT/2. Moreover,

for the interaction between two opposite charges separated by r, n = −1 and

〈2T 〉 = −〈V 〉.

27

Figure 1.3: Screen shot **of** using Mathematica to plot phase-plane for harmonic oscillator. Here

k/m = 1 and our xo = 0.75.

28

Chapter 2

Waves and Wavefunctions

In the world **of** quantum physics, no phenominon is a phenominon until it is a recorded

phenominon.

– John Archibald Wheler

The physical basis **of** quantum mechanics is

1. That matter, such as electrons, always arrives at a point as a discrete chunk, but that the

probibility **of** finding a chunk at a specified position is like the intensity distribution **of** a

wave.

2. The “quantum state” **of** a system is described by a mathematical object called a “wavefunction”

or state vector and is denoted |ψ〉.

3. The state |ψ〉 can be expanded in terms **of** the basis states **of** a given vector space, {|φi〉}

as

where 〈φi|ψ〉 denotes an inner product **of** the two vectors.

|ψ〉 = �

|φi〉〈φi|ψ〉 (2.1)

i

4. Observable quantities are associated with the expectation value **of** Hermitian operators and

that the eigenvalues **of** such operators are always real.

5. If two operators commute, one can measure the two associated physical quantities simultaneously

to arbitrary precision.

6. The result **of** a physical measurement projects |ψ〉 onto an eigenstate **of** the associated

operator |φn〉 yielding a measured value **of** an with probability |〈φn|ψ〉| 2 .

2.1 Position and Momentum Representation **of** |ψ〉

1 Two common operators which we shall use extensively are the position and momentum operator.

1 The majority **of** this lecture comes from Cohen-Tannoudji Chapter 1, part from Feynman & Hibbs

29

The position operator acts on the state |ψ〉 to give the amplitude **of** the system to be at a

given position:

ˆx|ψ〉 = |x〉〈x|ψ〉 (2.2)

= |x〉ψ(x) (2.3)

We shall call ψ(x) the wavefunction **of** the system since it is the amplitude **of** |ψ〉 at point x. Here

we can see that ψ(x) is an eigenstate **of** the position operator. We also define the momentum

operator ˆp as a derivative operator:

Thus,

ˆp = −i¯h ∂

∂x

(2.4)

ˆpψ(x) = −i¯hψ ′ (x). (2.5)

Note that ψ ′ (x) �= ψ(x), thus an eigenstate **of** the position operator is not also an eigenstate **of**

the momentum operator.

We can deduce this also from the fact that ˆx and ˆp do not commute. To see this, first consider

∂

∂x xf(x) = f(x) + xf ′ (x) (2.6)

Thus (using the shorthand ∂x as partial derivative with respect to x.)

[ˆx, ˆp]f(x) = i¯h(x∂xf(x) − ∂x(xf(x))) (2.7)

= −i¯h(xf ′ (x) − f(x) − xf ′ (x)) (2.8)

= i¯hf(x) (2.9)

What are the eigenstates **of** the ˆp operator? To find them, consider the following eigenvalue

equation:

ˆp|φ(k)〉 = k|φ(k)〉 (2.10)

Inserting a complete set **of** position states using the idempotent operator

�

I = |x〉〈x|dx (2.11)

and using the “coordinate” representation **of** the momentum operator, we get

Thus, the solution **of** this is (subject to normalization)

− i¯h∂xφ(k, x) = kφ(k, x) (2.12)

φ(k, x) = C exp(ik/¯h) = 〈x|φ(k)〉 (2.13)

30

We can also use the |φ(k)〉 = |k〉 states as a basis for the state |ψ〉 by writing

�

|ψ〉 = dk|k〉〈k|ψ〉 (2.14)

�

= dk|k〉ψ(k) (2.15)

where ψ(k) is related to ψ(x) via:

�

ψ(k) = 〈k|ψ〉 =

�

dx〈k|x〉〈x|ψ〉 (2.16)

= C dx exp(ikx/¯h)ψ(x). (2.17)

This type **of** integral is called a “Fourier Transfrom”. There are a number **of** ways to define

the normalization C when using this transform, for our purposes at the moment, we’ll set C =

1/ √ 2π¯h so that

and

ψ(x) = 1

�

√

2π¯h

ψ(x) = 1

�

√

2π¯h

dkψ(k) exp(−ikx/¯h) (2.18)

dxψ(x) exp(ikx/¯h). (2.19)

Using this choice **of** normalization, the transform and the inverse transform have symmetric forms

and we only need to remember the sign in the exponential.

2.2 The Schrödinger Equation

Postulate 2.1 The quantum state **of** the system is a solution **of** the Schrödinger equation

i¯h∂t|ψ(t)〉 = H|ψ(t)〉, (2.20)

where H is the quantum mechanical analogue **of** the classical Hamiltonian.

From classical mechanics, H is the sum **of** the kinetic and potential energy **of** a particle,

H = 1

2m p2 + V (x). (2.21)

Thus, using the quantum analogues **of** the classical x and p, the quantum H is

H = 1

2m ˆp2 + V (ˆx). (2.22)

To evaluate V (ˆx) we need a theorem that a function **of** an operator is the function evaluated

at the eigenvalue **of** the operator. The pro**of** is straight forward, Taylor expand the function

about some point, If

V (x) = (V (0) + xV ′ (0) + 1

2 V ′′ (0)x 2 · · ·) (2.23)

31

then

Since for any operator

Thus, we have

V (ˆx) = (V (0) + ˆxV ′ (0) + 1

2 V ′′ (0)ˆx 2 · · ·) (2.24)

So, in coordinate form, the Schrödinger Equation is written as

[ ˆ f, ˆ f p ] = 0∀ p (2.25)

〈x|V (ˆx)|ψ〉 = V (x)ψ(x) (2.26)

i¯h ∂

�

ψ(x, t) = −

∂t ¯h

2m

2.2.1 Gaussian Wavefunctions

∂ 2

+ V (x)

∂x2 �

ψ(x, t) (2.27)

Let’s assume that our initial state is a Gaussian in x with some initial momentum k◦.

ψ(x, 0) =

The momentum representation **of** this is

�

2

πa2 �1/4

exp(ikox) exp(−x 2 /a 2 ) (2.28)

ψ(k, 0) = 1

�

2π¯h

dxe −ikx ψ(x, 0) (2.29)

= (πa) 1/2 e −(k−ko)2a2 /4)

(2.30)

In Fig.2.1, we see a gaussian wavepacket centered about x = 0 with ko = 10 and a = 1.

For now we will use dimensionaless units. The red and blue components correspond to the real

and imaginary components **of** ψ and the black curve is |ψ(x)| 2 . Notice, that the wavefunction is

pretty localized along the x axis.

In the next figure, (Fig. 2.2) we have the momentum distribution **of** the wavefunction, ψ(k, 0).

Again, we have chosen ko = 10. Notice that the center **of** the distribution is shifted about ko.

So, for f(x) = exp(−x 2 /b 2 ), ∆x = b/ √ 2. Thus, when x varies form 0 to ±∆x, f(x) is

diminished by a factor **of** 1/ √ e. (∆x is the RMS deviation **of** f(x).)

For the Gaussian wavepacket:

or

Thus, ∆x∆p = ¯h/2 for the initial wavefunction.

∆x = a/2 (2.31)

∆k = 1/a (2.32)

∆p = ¯h/a (2.33)

32

0.75

0.5

0.25

-3 -2 -1 1 2 3

-0.25

-0.5

-0.75

Figure 2.1: Real (red), imaginary (blue) and absolute value (black) **of** gaussian wavepacket ψ(x)

è

y@kD

2.5

2

1.5

1

0.5

6 8 10 12 14

Figure 2.2: Momentum-space distribution **of** ψ(k).

33

k

2.2.2 Evolution **of** ψ(x)

Now, let’s consider the evolution **of** a free particle. By a “free” particle, we mean a particle

whose potential energy does not change, I.e. we set V (x) = 0 for all x and solve:

i¯h ∂

�

ψ(x, t) = −

∂t ¯h

2m

∂2 ∂x2 �

ψ(x, t) (2.34)

This equation is actually easier to solve in k-space. Taking the Fourier Transform,

Thus, the temporal solution **of** the equation is

i¯h∂tψ(k, t) = k2

ψ(k, t) (2.35)

2m

ψ(k, t) = exp(−ik 2 /(2m)t/¯h)ψ(k, 0). (2.36)

This is subject to some initial function ψ(k, 0). To get the coordinate x-representation **of** the

solution, we can use the FT relations above:

ψ(x, t) =

=

=

=

�

1

√ dkψ(k, t) exp(−ikx) (2.37)

2π¯h

�

dx ′ 〈x| exp(−iˆp 2 /(2m)t/¯h)|x ′ 〉ψ(x ′ , 0) (2.38)

�

�

�

m

2πi¯ht

�

dx ′ exp

� im(x − x ′ ) 2

2¯ht

ψ(x ′ , 0) (2.39)

dx ′ Go(x, x ′ )ψ(x ′ , 0) (2.40)

(homework: derive Go and show that Go is a solution **of** the free particle schrodinger equation

HGo = i∂tGo.) The function Go is called the “free particle propagator” or “Green’s Function”

and tells us the amplitude for a particle to start **of**f at x ′ and end up at another point x at time

t.

The sketch tells me that in order to got far away from the initial point in time t I need to

have a lot **of** energy (wiggles get closer together implies higher Fourier component )

Here we see that the probability to find a particle at the initial point decreases with time.

Since the period **of** oscillation (T ) is the time required to increase the phase by 2π.

2π = mx2 mx2

−

2¯ht 2¯h(t + T )

= mx2

2¯ht2 � �

2 T

1 + T/t

Let ω = 2π/T and take the long time limit t ≫ T , we can estimate

ω ≈ m

2¯h

� �

x 2

t

34

(2.41)

(2.42)

(2.43)

Figure 2.3: Go for fixed t as a function **of** x.

0.4

0.2

-10 -5 5 10

-0.2

-0.4

Since the classical kinetic energy is given by E = m/2v 2 , we obtain

E = ¯hω (2.44)

Thus, the energy **of** the wave is proportional to the period **of** oscillation.

We can evaluate the evolution in x using either the Go we derived above, or by taking the

FT **of** the wavefunction evolving in k-space. Recall that the solution in k-space was

ψ(k, t) = exp(−ik 2 /(2m)t/¯h)ψ(k, 0) (2.45)

Assuming a Gaussian form for ψ(k) as above,

√

a

ψ(x, t) =

(2π) 3/4

�

dke −a2 /4(k−ko) 2

e i(kx−ω(k)t)

where ω(k) is the dispersion relation for a free particle:

Cranking through the integral:

ψ(x, t) =

� 2a 2

π

� 1/4

�

ω(k) = ¯hk2

2m

e iφ

a 4 + 4¯h2 t 2

m 2

� 1/4 e ikox exp

where φ = −θ − ¯hk 2 o/(2m)t and tan 2θ = 2¯ht/(ma 2 ).

Likewise, for the amplitude:

�

(x − ¯hko/mt) 2

�

a 2 + 2i¯ht/m

|ψ(x, t)| 2 �

�

2

1

(x − v◦t)

=

exp −

2π∆x(t) 2 2∆x(t) 2

�

35

(2.46)

(2.47)

(2.48)

(2.49)

Figure 2.4: Evolution **of** a free particle wavefunction. In this case we have given the initial state

a kick in the +x direction. Notice that as the system moves, the center moves at a constant rate

where as the width **of** the packet constantly spreads out over time.

Where I define

∆x(t) = a

�

1 +

2

4¯h2 t2 m2a4 as the time dependent RMS width **of** the wave and the group velocity:

(2.50)

vo = ¯hko

. (2.51)

m

Now, since ∆p = ¯h∆k = ¯h/a is a constant for all time, the uncertainty relation becomes

∆x(t)∆p ≥ ¯h/2 (2.52)

corresponding to the particle’s wavefunction becoming more and more diffuse as it evolves in

time.

2.3 Particle in a Box

2.3.1 Infinite Box

The Mathematica handout shows how one can use Mathematica to set up and solve some simple

problems on the computer. (One good class problem would be to use Mathematica to carry

out the symbolic manipulations for a useful or interesting problem and/or to solve the problem

numerically.)

The potential we’ll work with for this example consists **of** two infinitely steep walls placed

at x = ℓ and x = 0 such that between the two walls, V (x) = 0. Within this region, we seek

solutions to the differential equation

∂ 2 xψ(x) = −2mE/¯h 2 ψ(x). (2.53)

The solutions **of** this are plane waves traveling to the left and to the right,

ψ(x) = A exp(−ikx) + B exp(+ikx) (2.54)

The coefficients A and B we’ll have to determine. k is determined by substitution back into the

differential equation

ψ ′′ (x) = −k 2 ψ(x) (2.55)

Thus, k 2 = 2mE/¯h 2 , or ¯hk = √ 2mE. Let’s work in units in which ¯h = 1 and me = 1. Energy in

these units is the Hartree (≈ 27.eV.) Posted on the web-page is a file (c-header file) which has a

number **of** useful conversion factors.

36

Since ψ(x) must vanish at x = 0 and x = ℓ

A + B = 0 (2.56)

A exp(ikℓ) + B exp(−ikℓ) = 0 (2.57)

We can see immediately that A = −B and that the solutions must correspond to a family **of** sine

functions:

Just a check,

ψ(x) = A sin(nπ/ℓx) (2.58)

ψ(ℓ) = A sin(nπ/ℓℓ) = A sin(nπ) = 0. (2.59)

To obtain the coefficient, we simply require that the wavefunctions be normalized over the range

x = [0, ℓ].

� ℓ

sin(nπx/ℓ)

0

2 dx = ℓ

(2.60)

2

Thus, the normalized solutions are

�

2

ψn(x) = sin(nπ/ℓx) (2.61)

ℓ

The eigenenergies are obtained by applying the Hamiltonian to the wavefunction solution

Thus we can write En as a function **of** n

Enψn(x) = − ¯h2

2m ∂2 xψn(x) (2.62)

= ¯h2 n 2 π 2

2a 2 m ψn(x) (2.63)

En = ¯h2 π2 2a2m n2

(2.64)

for n = 0, 1, 2, .... What about the case where n = 0? Clearly it’s an allowed solution **of**

the Schrödinger Equation. However, we also required that the probability to find the particle

anywhere must be 1. Thus, the n = 0 solution cannot be permitted.

Note also that the cosine functions are also allowed solutions. However, the restriction **of**

ψ(0) = 0 and ψ(ℓ) = 0 discounts these solutions.

In Fig. 2.5 we show the first few eigenstates for an electron trapped in a well **of** length a = π.

The potential is shown in gray. Notice that the number **of** nodes increases as the energy increases.

In fact, one can determine the state **of** the system by simply counting nodes.

What about orthonormality. We stated that the solution **of** the eigenvalue problem form an

orthonormal basis. In Dirac notation we can write

�

〈ψn|ψm〉 = dx〈ψn|x〉〈x|ψm〉 (2.65)

=

� ℓ

0

� ℓ

dxψ ∗ n(x)ψm(x) (2.66)

= 2

ℓ 0

dx sin(nπx/ℓ) sin(mπx/ℓ) (2.67)

= δnm. (2.68)

37

14

12

10

8

6

4

2

-1 1 2 3 4

Figure 2.5: Particle in a box states

Thus, we can see in fact that these solutions do form a complete set **of** orthogonal states on

the range x = [0, ℓ]. Note that it’s important to specify “on the range...” since clearly the sin

functions are not a set **of** orthogonal functions over the entire x axis.

2.3.2 Particle in a finite Box

Now, suppose our box is finite. That is

V (x) =

�

−Vo if −a < x < a

0 otherwise

(2.69)

Let’s consider the case for E < 0. The case E > 0 will correspond to scattering solutions. In

side the well, the wavefunction oscillates, much like in the previous case.

ψW (x) = A sin(kix) + B cos(kix) (2.70)

where ki comes from the equation for the momentum inside the well

¯hki =

�

2m(En + Vo) (2.71)

We actually have two classes **of** solution, a symmetric solution when A = 0 and an antisymmetric

solution when B = 0.

Outside the well the potential is 0 and we have the solutions

ψO(x) = c1e ρx andc2e −ρx

38

(2.72)

We will choose the coefficients c1 and c2 as to create two cases, ψL and ψR on the left and right

hand sides **of** the well. Also,

¯hρ = √ −2mE (2.73)

Thus, we have three pieces **of** the full solution which we must hook together.

ψL(x) = Ce ρx for x < −a (2.74)

ψR(x) = De −ρx for x > −a (2.75)

(2.76)

ψW (x) = A sin(kix) + B cos(kix)for inside the well (2.77)

To find the coefficients, we need to set up a series **of** simultaneous equations by applying the

conditions that a.) the wavefunction be a continuous function **of** x and that b.) it have continuous

first derivatives with respect to x. Thus, applying the two conditions at the boundaries:

The matching conditions at x = a

The final results are (after the chalk dust settles):

ψL(−a) − ψW (−a) = 0 (2.78)

(2.79)

ψR(a) − ψW (a) = 0 (2.80)

(2.81)

ψ ′ L(−a) − ψ ′ W (−a) = 0 (2.82)

(2.83)

ψ ′ R(a) − ψ ′ W (a) = 0 (2.84)

1. For A = 0. B = D sec(aki)e −aρ and C = D. (Symmetric Solution)

2. For B = 0, A = C csc(aki)e −aρ and C = −D. (Antisymmetric Solution)

So, now we have all the coefficients expressed in terms **of** D, which we can determine by normalization

(if so inclined). We’ll not do that integral, as it is pretty straightforward.

For the energies, we substitute the symmetric and antisymmetric solutions into the Eigenvalue

equation and obtain:

ρ cos(aki) = ki sin(ki) (2.85)

39

or

for the symmetric case and

�

for the anti-symmetric case, or

�

E

Vo − E

ρ

ki

= tan(aki) (2.86)

= tan(a

�

2m(Vo − E)/¯h) (2.87)

ρ sin(aki) = −ki cos(aki) (2.88)

E

Vo − E

ρ

ki

= cot(aki) (2.89)

(2.90)

= cot(a

�

2m(Vo − E)/¯h) (2.91)

Substituting the expressions for ki and ρ into final results for each case we find a set **of**

matching conditions: For the symmetric case, eigenvalues occur when ever the two curves

and for the anti-symmetric case,

�

�

1 − Vo/E = tan(a 2m(E − Vo)/¯h) (2.92)

�

�

1 − Vo/E = cot(a 2m(E − Vo)/¯h) (2.93)

These are called “transcendental” equations and closed form solutions are generally impossible

to obtain. Graphical solutions are helpful. In Fig. ?? we show the graphical solution to the

transendental equations for an electron in a Vo = −10 well **of** width a = 2. The black dots

indicate the presence **of** two bound states, one symmetric and one anti-symmetric at E = 2.03

and 3.78 repectively.

2.3.3 Scattering states and resonances.

Now let’s take the same example as above, except look at states for which E > 0. In this case, we

have to consider where the particles are coming from and where they are going. We will assume

that the particles are emitted with precise energy E towards the well from −∞ and travel from

left to right. As in the case above we have three distinct regions,

1. x > −a where ψ(x) = e ik1x + Re −ik1x = ψL(x)

2. −a ≤ x ≤ +a where ψ(x) = Ae −ik2x + Be +ik2x = ψW (x)

40

symêasym

4

3

2

1

-1

-2

-3

-4

2 4 6 8 10 E

Figure 2.6: Graphical solution to transendental equations for an electron in a truncated hard

well **of** depth Vo = 10 and width a = 2. The short-dashed blue curve corresponds to the

symmetric � case and the long-dashed blue curve corresponds to the asymetric case. The red line

is 1 − V o/E. Bound state solution are such that the red and blue curves cross.

3. x > +a where ψ(x) = T e +ik1x = ψR(x)

where k1 = √ �

2mE/¯h is the momentum outside the well, k2 = 2m(E − V )/¯h is the momentum

inside the well, and A, B, T , and R are coefficients we need to determine. We also have the

matching conditions:

ψL(−a) − ψW (−a) = 0

ψ ′ L(−a) − ψ ′ W (−a) = 0

ψR(a) − ψW (a) = 0

ψ ′ R(a) − ψ ′ W (a) = 0

This can be solved by hand, however, Mathematica make it easy. The results are a series **of** rules

which we can use to determine the transmission and reflection coefficients.

T →

A →

B →

R →

−4e−2iak1+2iak2k1k2 −k1 2 + e4iak2k1 2 − 2k1k2 − 2e4iak2k1k2 − k2 2 + e4iak2k2 2 ,

2e−iak1+3iak2k1 (k1 − k2)

−k1 2 + e4iak2k1 2 − 2k1k2 − 2e4iak2k1k2 − k2 2 + e4iak2k2 2 ,

−2e−iak1+iak2k1 (k1 + k2)

−k1 2 + e4iak2k1 2 − 2k1k2 − 2e4iak2k1k2 − k2 2 + e4iak2k2 2 ,

�

−1 + e4iak2 � �

k1 2 − k2 2�

e2iak1 �

−k1 2 + e4iak2k1 2 − 2k1k2 − 2e4iak2k1k2 − k2 2 + e4iak2k2 2�

41

1

0.8

0.6

0.4

0.2

R,T

10 20 30 40

En HhartreeL

Figure 2.7: Transmission (blue) and Reflection (red) coefficients for an electron scattering over

a square well (V = −40 and a = 1 ).

The R and T coefficients are related to the rations **of** the reflected and transimitted flux to

the incoming flux. The current operator is given by

Inserting the wavefunctions above yields:

j(x) = ¯h

2mi (ψ∗ ∇ψ − ψ∇ψ ∗ ) (2.94)

jin = ¯hk1

m

2 ¯hk1R

jref = −

m

2 ¯hk1T

jtrans =

m

Thus, R 2 = −jref/jin and T 2 = jtrans/jin. In Fig. 2.7 we show the transmitted and reflection

coefficients for an electron passing over a well **of** depth V = −40, a = 1 as a function **of** incident

energy, E.

Notice that the transmission and reflection coefficients under go a series oscillations as the

incident energy is increased. These are due to resonance states which lie in the continuum. The

condition for these states is such that an integer number **of** de Broglie wavelength **of** the wave in

the well matches the total length **of** the well.

λ/2 = na

Fig. 2.8,show the transmission coefficient as a function **of** both incident energy and the well

depth and (or height) over a wide range indicating that resonances can occur for both wells and

bumps. Figures 2.9 show various scattering wavefunctions for on an **of**f-resonance cases. Lastly,

Fig. ?? shows an Argand plot **of** both complex components **of** ψ.

42

1

0.95

T

0.9

0.85

10

En

20

30

-10

40

Figure 2.8: Transmission Coefficient for particle passing over a bump. Here we have plotted T as

a function **of** V and incident energy En. The oscillations correspond to resonance states which

occur as the particle passes over the well (for V < 0) or bump V > 0.

2.3.4 Application: **Quantum** Dots

One **of** the most active areas **of** research in s**of**t condensed matter is that **of** designing physical

systems which can confine a quantum state in some controllable way. The idea **of** engineering a

quantum state is extremely appealing and has numerous technological applications from small

logic gates in computers to optically active materials for biomedical applications. The basic

physics **of** these materials is relatively simple and we can use the basic ideas presented in this

chapter. The basic idea is to layer a series **of** materials such that electrons can be trapped in a

geometrically confined region. This can be accomplished by insulator-metal-insulator layers and

etching, creating disclinations in semiconductors, growing semi-conductor or metal clusters, and

so on. A quantum dot can even be a defect site.

We will assume through out that our quantum well contains a single electron so that we can

treat the system as simply as possible. For a square or cubic quantum well, energy levels are

simply those **of** an n-dimensional particle in a box. For example for a three dimensional system,

Enx,ny,nz = ¯h2 π2 ⎛

⎝

2m

� �

nx

2

Lx

+

-5

� �2 ny

Ly

+

0

� nz

Lz

5

V

10

⎞

�2 ⎠ (2.95)

where Lx, Ly, and Lz are the lengths **of** the box and m is the mass **of** an electron.

The density **of** states is the number **of** energy levels per unit energy. If we take the box to be

43

1.5

1

0.5

-10 -5 5 10

-0.5

-1

-1.5

1

0.5

-10 -5 5 10

-0.5

-1

Figure 2.9: Scattering waves for particle passing over a well. In the top graphic, the particle is

partially reflected from the well (V < 0) and in the bottom graphic, the particle passes over the

well with a slightly different energy than above, this time with little reflection.

a cube Lx = Ly = Lz we can relate n to a radius **of** a sphere and write the density **of** states as

ρ(n) = 4π 2 2 dn

n

dE = 4π2n 2

Thus, for a 3D cube, the density **of** states is

�

2 4mL

ρ(n) =

π¯h 2

�

n

� �−1 dE

.

dn

i.e. for a three dimensional cube, the density **of** states increases as n and hence as E 1/2 .

Note that the scaling **of** the density **of** states with energy depends strongly upon the dimen-

44

Im@yD

4

2

0

-2

-4

-10

x

0

10

-5

0

Figure 2.10: Argand plot **of** a scattering wavefunction passing over a well. (Same parameters as

in the top figure in Fig. 2.9).

5

Re@yD

sionality **of** the system. For example in one dimension,

and in two dimensions

ρ(n) = 2mL2

¯h 2 π 2

ρ(n) = const.

The reason for this lies in the way the volume element for linear, circular, and spherical integration

scales with radius n. Thus, measuring the density **of** states tells us not only the size **of** the system,

but also its dimensionality.

We can generalize the results here by realizing that the volume **of** a d dimensional sphere in

k space is given by

Vd = kd π d/2

Γ(1 + d/2)

where Γ(x) is the gamma-function. The total number **of** states per unit volume in a d-dimensional

space is then

nk = 2 1

Vd

2π2 and the density is then the number **of** states per unit energy. The relation between energy and

k is

Ek = ¯h2

2m k2 .

i.e.

√

2Ekm

k =

¯h

45

1

n

3

2.5

2

1.5

1

0.5

DOS

which gives

0.2 0.4 0.6 0.8 1

energy HauL

Figure 2.11: Density **of** states for a 1-, 2- , and 3- dimensional space.

d

d

−2+

2−1+ 2 d π 2

� √ �

m ɛ

d

¯h

ρd(E) =

ɛ Γ(1 + d

2 )

A quantum well is typically constructed so that the system is confined in one dimension

and unconfined in the other two. Thus, a quantum well will typically have discrete state only

in the confined direction. The density **of** states for this system will be identical to that **of** the

3-dimensional system at energies where the k vectors coincide. If we take the thickness to be s,

then the density **of** states for the quantum well is

ρ = L

s ρ2(E)

�

L ρ3(E)

�

Lρ2(E)/s

where ⌊x⌋ is the ”floor” function which means take the largest integer less than x. This is plotted

in Fig. 2.12 and the stair-step DOS is indicative **of** the embedded confined structure.

Next, we consider a quantum wire **of** thickness s along each **of** its 2 confined directions. The

DOS along the unconfined direction is one-dimensional. As above, the total DOS will be identical

46

30

20

10

DOS **Quantum** well vs. 3d body

e HauL

0.005 0.01 0.015 0.02 0.025 0.03

120

100

DOS **Quantum** wire vs. 3d body

80

60

40

20

e HauL

0.05 0.1 0.15 0.2 0.25 0.3

Figure 2.12: Density **of** states for a quantum well and quantum wire compared to a 3d space.

Here L = 5 and s = 2 for comparison.

to the 3D case when the wavevectors coincide. Increasing the radius **of** the wire eventually leads

to the case where the steps decrease and merge into the 3D curve.

� �2

�

2 L L ρ2(E)

ρ = ρ1(E)

s L2 �

ρ2(E)/s

For a spherical dot, we consider the case in which the radius **of** the quantum dot is small

enough to support discrete rather than continuous energy levels. In a later chapter, we will derive

this result in more detail, for now we consider just the results. First, an electron in a spherical

dot obeys the Schrödinger equation:

where ∇ 2 is the Laplacian operator in spherical coordinates

∇ 2 = 1

r

∂ 2

r +

∂r2 1

− ¯h2

2m ∇2 ψ = Eψ (2.96)

r 2 sin θ

∂ ∂

sin θ

∂θ ∂θ +

1

r 2 sin 2 θ

∂2 .

∂φ2 The solution **of** the Schrodinger equation is subject to the boundary condition that for r ≥ R,

ψ(r) = 0, where R is the radius **of** the sphere and are given in terms **of** the spherical Bessel

function, jl(r) and spherical harmonic functions, Ylm.

with energy

ψnlm = 21/2

R3/2 jl(αr/R)

jl+1(α) Ylm(Ω), (2.97)

E = ¯h2 α

2m

2

R2 (2.98)

Note that the spherical Bessel functions (**of** the first kind) are related to the Bessel functions via,

jl(x) =

� π

2x Jl+1/2(x). (2.99)

47

1

0.8

0.6

0.4

0.2

-0.2

j lHxL

5 10 15 20 x

Figure 2.13: Spherical Bessel functions, j0, j1, and j1 (red, blue, green)

The first few **of** these are

sin x

j0(x) = (2.100)

x

sin x cos x

j1(x) = − (2.101)

x2 x

� �

3 1

j2(x) = − sin x −

x3 x

3

cos x (2.102)

x2 jn(x) = (−1) n x n

� �n 1 d

j0(x) (2.103)

x dx

where the last line provides a way to generate jn from j0.

The α’s appearing in the wavefunction and in the energy expression are determined by the

boundary condition that ψ(R) = 0. Thus, for the lowest energy state we require

i.e. α = π. For the next state (l = 1),

j1(α) =

j0(α) = 0, (2.104)

sin α cos α

−

α2 α

= 0. (2.105)

This can be solved to give α = 4.4934. These correspond to where the spherical Bessel functions

pass through zero. The first 6 **of** these are 3.14159, 4.49341, 5.76346, 6.98793, 8.18256, 9.35581.

These correspond to where the first zeros occur and give the condition for the radial quantization,

n = 1 with angular momentum l = 0, 1, 2, 3, 4, 5. There are more zeros, and these correspond to

the case where n > 1.

In the next set **of** figures (Fig. 2.14), we look at the radial wavefunctions for an electron in

a 0.5˚Aquantum dot. First, the case where n = 1, l = 0 and n = 0, l = 1. In both cases, the

wavefunctions vanish at the radius **of** the dot. The radial probability distribution function (PDF)

is given by P = r 2 |ψnl(r)| 2 . Note that increasing the angular momentum l from 0 to 1 causes

the electron’s most probable position to shift outwards. This is due to the centrifugal force due

to the angular motion **of** the electron. For the n, l = (2, 0) and (2, 1) states, we have 1 node in

the system and two peaks in the PDF functions.

48

12

10

5

-5

-10

-15

-20

-25

8

6

4

2

y

y

0.1 0.2 0.3 0.4 0.5 r

0.1 0.2 0.3 0.4 0.5 r

4

3

2

1

P

4

3

2

1

P

0.1 0.2 0.3 0.4 0.5 r

0.1 0.2 0.3 0.4 0.5 r

Figure 2.14: Radial wavefuncitons (left column) and corresponding PDFs (right column) for an

electron in a R = 0.5˚Aquantum dot. The upper two correspond to (n, l) = (1, 0) (solid) and

(n, l) = (1, 1) (dashed) while the lower correspond to (n, l) = (2, 0) (solid) and (n, l) = (2, 1)

(dashed) .

2.4 Tunneling and transmission in a 1D chain

In this example, we are going to generalize the ideas presented here and look at what happens if

we discretize the space in which a particle can move. This happens physically when we consider

what happens when a particle (eg. an electron) can hop from one site to another. If an electron

is on a given site, it has a certain energy ε to be there and it takes energy β to move the electron

from the site to its neighboring site. We can write the Schrödinger equation for this system as

ujε + βuj+1 + βuj−1 = Euj.

for the case where the energy depends upon where the electron is located. If the chain is infinite,

we can write uj = T e ikdj and find that the energy band goes as E = ε + 2β cos(kd) where k is

now the momentum **of** the electron.

2.5 Summary

We’ve covered a lot **of** ground. We now have enough tools at hand to begin to study some physical

systems. The traditional approach to studying quantum mechanics is to progressively a series

**of** differential equations related to physical systems (harmonic oscillators, angular momentum,

49

hydrogen atom, etc...). We will return to those models in a week or so. Next week, we’re going to

look at 2 and 3 level systems using both time dependent and time-independent methods. We’ll

develop a perturbative approach for computing the transition amplitude between states. We will

also look at the decay **of** a state when its coupled to a continuum. These are useful models for

a wide variety **of** phenomena. After this, we will move on to the harmonic oscillator.

2.6 Problems and Exercises

Exercise 2.1 1. Derive the expression for

where ho is the free particle Hamiltonian,

Go(x, x ′ ) = 〈x| exp(−ihot/¯h)|x ′ 〉 (2.106)

ho = − ¯h2 ∂

2m

2

∂x2 2. Show that Go is a solution **of** the free particle Schrödinger Equation

(2.107)

i¯h∂tGo(t) = hoGo(t). (2.108)

Exercise 2.2 Show that the normalization **of** a wavefunction is independent **of** time.

Solution:

i∂t〈ψ(t)|ψ(t)〉 = (i〈 ˙ ψ(t)|)(|ψ(t)〉) + (〈ψ(t)|)(i| ˙ ψ(t)〉) (2.109)

= −〈ψ(t)| ˆ H † |ψ(t)〉 + 〈ψ(t)| ˆ H|ψ(t)〉 (2.110)

= −〈ψ(t)| ˆ H|ψ(t)〉 + 〈ψ(t)| ˆ H|ψ(t)〉 = 0 (2.111)

Exercise 2.3 Compute the bound state solutions (E < 0) for a square well **of** depth Vo where

�

−Vo

V (x) =

0

−a/2 ≤ x ≤ a/2

otherwise

(2.112)

1. How many energy levels are supported by a well **of** width a.

2. Show that a very narrow well can support only 1 bound state, and that this state is an even

function **of** x.

3. Show that the energy **of** the lowest bound state is

4. Show that as

ρ =

E ≈

mV 2

o a 2

2¯h 2

the probability **of** finding the particle inside the well vanishes.

�

(2.113)

− 2mE

2 → 0 (2.114)

¯h

50

Exercise 2.4 Consider a particle with the potential

V (x) =

⎧

⎪⎨

⎪⎩

0 for x > a

−Vo for 0 ≤ x ≤ a

∞ for x < 0

(2.115)

1. Let φ(x) be a stationary state. Show that φ(x) can be extended to give an odd wavefunction

corresponding to a stationary state **of** the symmetric well **of** width 2a (i.e the one studied

above) and depth Vo.

2. Discuss with respect to a and Vo the number **of** bound states and argue that there is always

at least one such state.

3. Now turn your attention toward the E > 0 states **of** the well. Show that the transmission

**of** the particle into the well region vanishes as E → 0 and that the wavefunction is perfectly

reflected **of**f the sudden change in potential at x = a.

Exercise 2.5 Which **of** the following are eigenfunctions **of** the kinetic energy operator

e x , x 2 , x n ,3 cos(2x), sin(x) + cos(x), e −ikx ,

.

Solution Going in order:

1. e x

2. x 2

3. x n

4. 3 cos(2x)

5. sin(x) + cos(x)

6. e −ikx

ˆT = − ¯h2 ∂

2m

2

∂x2 f(x − x ′ � ∞

) = dke −ik(x−x′ ) −ik

e 2 /(2m)

−∞

(2.116)

(2.117)

Exercise 2.6 Which **of** the following would be acceptable one dimensional wavefunctions for a

bound particle (upon normalization): f(x) = e−x , f(x) = e−x2, f(x) = xe−x2, and

Solution In order:

f(x) =

�

e−x2 2e−x2 51

x ≥ 0

x < 0

(2.118)

1. f(x) = e −x

2. f(x) = e −x2

3. f(x) = xe −x2

4.

f(x) =

�

e−x2 2e−x2 x ≥ 0

x < 0

Exercise 2.7 For a one dimensional problem, consider a particle with wavefunction

ψ(x) = N exp(ipox/¯h)

√ x 2 + a 2

where a and po are real constants and N the normalization.

1. Determine N so that ψ(x) is normalized.

� ∞

−∞

Thus ψ(x) is normalized when

dx|ψ(x)| 2 = N 2

� ∞

N =

2 π

= N

a

� a

π

−∞

dx

1

x 2 + a 2

(2.119)

(2.120)

(2.121)

(2.122)

(2.123)

2. The position **of** the particle is measured. What is the probability **of** finding a result between

−a/ √ 3 and +a/ √ 3?

� √

a +a/ 3

π −a/ √ dx|ψ(x)|

3

2 =

� +a/ √ 3

−a/ √ 3

dx

= 1

π tan−1 (x/a)

= 1

3

1

x2 + a2 �

�+a/

�

�

√ 3

−a/ √ 3

3. Compute the mean value **of** a particle which has ψ(x) as its wavefunction.

(2.124)

(2.125)

(2.126)

〈x〉 = a

� ∞ x

dx

π −∞ x2 + a2 (2.127)

= 0 (2.128)

52

Exercise 2.8 Consider the Hamiltonian **of** a particle in a 1 dimensional well given by

H = 1

2m ˆp2 + ˆx 2

where ˆx and ˆp are position and momentum operators. Let |φn〉 be a solution **of**

for n = 0, 1, 2, · · ·. Show that

(2.129)

H|φn〉 = En|φn〉 (2.130)

〈φn|ˆp|φm〉 = αnm〈φn|ˆx|φm〉 (2.131)

where αnm is a coefficient depending upon En − Em. Compute αnm. (Hint: you will need to use

the commutation relations **of** [ˆx, H] and [ˆp, H] to get this). Finally, from all this, deduce that

�

m

(En − Em) 2 |φn|ˆx|φm〉| 2 = ¯h2

2m 〈φn|ˆp 2 |φn〉 (2.132)

Exercise 2.9 The state space **of** a certain physical system is three-dimensional. Let |u1〉, |u2〉,

and |u3〉 be an orthonormal basis **of** the space in which the kets |ψ1〉 and |ψ2〉 are defined by

1. Are the states normalized?

|ψ1〉 = 1

√ 2 |u1〉 + i

2 |u2〉 + 1

2 |u3〉 (2.133)

|ψ2〉 = 1

√ 3 |u1〉 + i

√ 3 |u3〉 (2.134)

2. Calculate the matrices, ρ1 and ρ2 representing in the {|ui〉〉 basis, the projection operators

onto |ψ1〉 and |ψ2〉. Verify that these matrices are Hermitian.

Exercise 2.10 Let ψ(r) = ψ(x, y, z) be the normalized wavefunction **of** a particle. Express in

terms **of** ψ(r):

1. A measurement along the x-axis to yield a result between x1 and x2.

2. A measurement **of** momentum component px to yield a result between p1 and p2.

3. Simultaneous measurements **of** x and pz to yield x1 ≤ x ≤ x2 and pz > 0.

4. Simultaneous measurements **of** px, py, and pz, to yield

p1 ≤ px ≤ p2

p3 ≤ py ≤ p4

p5 ≤ pz ≤ p6

(2.135)

(2.136)

(2.137)

(2.138)

(2.139)

Show that this result is equal to the result **of** part 2 when p3, p5 → −∞ and p4, p6 → +∞.

53

Exercise 2.11 Consider a particle **of** mass m whose potential energy is

V (x) = −α(δ(x + l/2) + δ(x − l/2))

1. Calculate the bound states **of** the particle, setting

Show that the possible energies are given by

E = − ¯h2 ρ 2

2m .

e −ρl �

= ± 1 − 2ρ

�

µ

where µ = 2mα/¯h 2 . Give a graphic solution **of** this equation.

(a) The Ground State. Show that the ground state is even about the origin and that it’s

energy, Es is less than the bound state **of** a particle in a single δ-function potential,

−EL. Interpret this physically. Plot the corresponding wavefunction.

(b) Excited State. Show that when l is greater than some value (which you need to determine),

there exists an odd excited state **of** energy EA with energy greater than −EL.

Determine and plot the corresponding wavefunction.

(c) Explain how the preceeding calculations enable us to construct a model for an ionized

diatomic molecule, eg. H + 2 , whose nuclei are separated by l. Plot the energies **of** the

two states as functions **of** l, what happens as l → ∞ and l → 0?

(d) If we take Coulombic repulsion **of** the nuclei into account, what is the total energy

**of** the system? Show that a curve which gives the variation with respect to l **of** the

energies thus obtained enables us to predict in certain cases the existence **of** bound

states **of** H + 2 and to determine the equilibrium bond length.

2. Calculate the reflection and transmission coefficients for this system. Plot R and T as

functions **of** l. Show that resonances occur when l is an integer multiple **of** the de Broglie

wavelength **of** the particle. Why?

54

Chapter 3

Semi-Classical **Quantum** **Mechanics**

Good actions ennoble us, and we are the sons **of** our own deeds.

–Miguel de Cervantes

The use **of** classical mechanical analogs for quantum behavour holds a long and proud tradition in

the development and application **of** quantum theory. In Bohr’s original formulation **of** quantum

mechanics to explain the spectra **of** the hydrogen atom, Bohr used purely classical mechanical

notions **of** angular momentum and rotation for the basic theory and imposed a quantization

condition that the angular momentum should come in integer multiples **of** ¯h. Bohr worked under

the assumption that at some point the laws **of** quantum mechanics which govern atoms and

molecules should correspond to the classical mechanical laws **of** ordinary objects like rocks and

stones. Bohr’s Principle **of** Correspondence states that quantum mechanics was not completely

separate from classical mechanics; rather, it incorporates classical theory.

From a computational viewpoint, this is an extremely powerful notion since performing a

classical trajectory calculation (even running 1000’s **of** them) is simpler than a single quantum

calculation **of** a similar dimension. Consequently, the development **of** semi-classical methods has

and remains an important part **of** the development and untilization **of** quantum theory. In fact

even in the most recent issues **of** the Journal **of** Chemical Physics, Phys. Rev. Lett, and other

leading physics and chemical physics journals, one finds new developments and applications **of**

this very old idea.

In this chapter we will explore this idea in some detail. The field **of** semi-classical mechanics

is vast and I would recommend the following for more information:

1. Chaos in Classical and **Quantum** **Mechanics**, Martin Gutzwiller (Springer-Verlag, 1990).

Chaos in quantum mechanics is a touchy subject and really has no clear-cut definition that

anyone seems to agree upon. Gutzwiller is one **of** the key figures in sorting all this out. This

is very nice and not too technical monograph on quantum and classical correspondence.

2. Semiclassical Physics, M. Brack and R. Bhaduri (Addison-Wesley, 1997). Very interesting

book, mostly focusing upon many-body applications and Thomas-Fermi approximations.

3. Computer Simulations **of** Liquids, M. P. Allen and D. J. Tildesley (Oxford, 1994). This

book mostly focus upon classical MD methods, but has a nice chapter on quantum methods

which were state **of** the art in 1994. Methods come and methods go.

There are many others, **of** course. These are just the ones on my bookshelf.

55

3.1 Bohr-Sommerfield quantization

Let’s first review Bohr’s original derivation **of** the hydrogen atom. We will go through this a bit

differently than Bohr since we already know part **of** the answer. In the chapter on the Hydrogen

atom we derived the energy levels in terms **of** the principle quantum number, n.

En = − me4

2¯h 2

In Bohr’s correspondence principle, the quantum energy must equal the classical energy. So

for an electron moving about a proton, that energy is inversely proportional to the distance **of**

separation. So, we can write

− me4

2¯h 2

1

n 2

1 e2

= −

n2 2r

Now we need to figure out how angular momentum gets pulled into this. For an orbiting body

the centrifugal force which pulls the body outward is counterbalenced by the inward tugs **of** the

centripetal force coming from the attractive Coulomb potential. Thus,

(3.1)

(3.2)

mrω 2 = e2

, (3.3)

r2 where ω is the angular frequency **of** the rotation. Rearranging this a bit, we can plug this into

the RHS **of** Eq. 3.2 and write

− me4

2¯h 2

1

n2 = −mr3 ω2 2r

The numerator now looks amost like the classical definition **of** angular momentum: L = mr 2 ω.

So we can write the last equation as

Solving for L 2 :

− me4

2¯h 2

(3.4)

1 L2

= − . (3.5)

n2 2mr2 L 2 = me4

2¯h 2

2mr2 n2 . (3.6)

Now, we need to pull in another one **of** Bohr’s results for the orbital radius **of** the H-atom:

Plug this into Eq.3.6 and after the dust settles, we find

r = ¯h2

me 2 n2 . (3.7)

L = ¯hn. (3.8)

But, why should electrons be confined to circular orbits? Eq. 3.8 should be applicable to

any closed path the electron should choose to take. If the quantization condition only holds

56

for circular orbits, then the theory itself is in deep trouble. At least that’s what Sommerfield

thought.

The numerical units **of** ¯h are energy times time. That is the unit **of** action in classical

mechanics. In classical mechanics, the action **of** a mechanical system is given by the integral **of**

the classical momentum along a classical path:

S =

� x 2

x1

pdx (3.9)

For an orbit, the initial point and the final point must coincide, x1 = x2, so the action integral

must describe some the area circumscribed by a closed loop on the p−x plane called phase-space.

�

S = pdx (3.10)

So, Bohr and Sommerfield’s idea was that the circumscribed area in phase-space was quantized

as well.

As a check, let us consider the harmonic oscillator. The classical energy is given by

E(p, q) = p2 k

+

2m 2 q2 .

This is the equation for an ellipse in phase space since we can re-arrange this to read

1 =

p 2

2mE

= p2 q2

+

a2 b2 + k

2E q2

(3.11)

where a = √ �

2mE and b = 2E/k describe the major and minor axes **of** the ellipse. The area **of**

an ellipse is A = πab, so the area circumscribed by a classical trajectory with energy E is

�

S(E) = 2Eπ m/k (3.12)

Since

�

k/m = ω, S = 2πE/ω = E/ν. Finally, since E/ν must be an integer multiple **of** h, the

Bohr-Sommerfield condition for quantization becomes

�

pdx = nh (3.13)

�

where p is the classical momentum for a path **of** energy E, p = 2m(V (x) − E. Taking this a

bit farther, the de Broglie wavelength is p/h, so the Bohr-Sommerfield rule basically states that

stationary energies correspond to classical paths for which there are an integer number **of** de

Broglie wavelengths.

Now, perhaps you can see where the problem with quantum chaos. In classical chaos, chaotic

trajectories never return to their exact staring point in phase-space. They may come close, but

there are no closed orbits. For 1D systems, this is does not occur since the trajectories are the

contours **of** the energy function. For higher dimensions, the dimensionality **of** the system makes

it possible to have extremely complex trajectories which never return to their starting point.

Exercise 3.1 Apply the Bohr-Sommerfield proceedure to determine the stationary energies for

a particle in a box **of** length l.

57

3.2 The WKB Approximation

The original Bohr-Sommerfield idea can be imporoved upon considerably to produce an asymptotic

(¯h → 0) approximation to the Schrödinger wave function. The idea was put forward at about

the same time by three different theoreticians, Brillouin (in Belgium), Kramers (in Netherlands),

and Wentzel (in Germany). Depending upn your point **of** origin, this method is the WKB (US

& Germany), BWK (France, Belgium), JWKB (UK), you get the idea. The original references

are

1. “La mécanique odularatoire de Schrödinger; une méthode générale de résolution par approximations

successives”, L. Brillouin, Comptes rendus (Paris). 183, 24 (1926).

2. “Wellenmechanik und halbzahlige Quantisierung”, H. A. Kramers, Zeitschrift für Physik

39, 828 (1926).

3. “Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik”,

Zeitschrift für Physik 38, 518 (1926).

We will first go through how one can use the approach to determine the eigenvalues **of** the

Schrödinger equation via semi-classical methods, then show how one can approximate the actual

wavefunctions themselves.

3.2.1 Asymptotic expansion for eigenvalue spectrum

The WKB proceedure is initiated by writing the solution to the Schödinger equation

as

ψ ′′ + 2m

2 (E − V (x))ψ = 0

¯h

� �

i

ψ(x) = exp

¯h

�

χdx . (3.14)

We will soon discover that χ is the classical momentum **of** the system, but for now, let’s consider

it to be a function **of** the energy **of** the system. Substituting into the Schrodinger equation

produces a new differential equation for χ

If we take ¯h → 0, it follows then that

¯h dχ

i dx = 2m(E − V ) − χ2 . (3.15)

χ = χo =

�

2m(E − V ) = |p| (3.16)

which is the magnitude **of** the classical momentum **of** a particle. So, if we assume that this is

simply the leading order term in a series expansion in ¯h we would have

χ = χo + ¯h

i χ1 +

58

� �2 ¯h

χ2 . . . (3.17)

i

Substituting Eq. 3.17 into

χ = ¯h

i

1

ψ

∂ψ

x

(3.18)

and equating to zero coefficients with different powers **of** ¯h, one obtains equations which determine

the χn corrections in succession:

for n = 1, 2, 3 . . .. For example,

and so forth.

χ2 = − χ2 1 + χ ′ 1

2χo

= − 1

�

2χo

= −

d

dx χn−1

n�

= − χn−mχm

m=0

χ1 = − 1

2

χ ′ o

χo

V ′2

+

16(E − V ) 2

= 1 V

4

′

E − V

V ′2

+

4(E − V ) 2

V ′′

�

.

4(E − V )

5V ′2

32(2m) 1/2 V

−

(E − V ) 5/2 ′′

8(2m) 1/2 (E − V ) 3/2

Exercise 3.2 Verify Eq. 3.19 and derive the first order correction in Eq.3.20.

(3.19)

(3.20)

(3.21)

Now, to use these equations to determine the spectrum, we replace x everywhere by a complex

coordinate z and suppose that V (z) is a regular and analytic function **of** z in any physically

relevant region. Consequently, we can then say that ψ(z) is an analytic function **of** z. So, we

can write the phase integral as

n = 1

�

h

= 1

2πi

C

�

χ(z)dz

C

ψ ′ n(z)

dz (3.22)

ψn(z)

where ψn is the nth discrete stationary solution to the Schrödinger equation and C is some

contour **of** integration on the z plane. If there is a discrete spectrum, we know that the number

**of** zeros, n, in the wavefunction is related to the quantum number corresponding to the n + 1

energy level. So if ψ has no real zeros, this is the ground state wavefunction with energy Eo, one

real zero corresponds to energy level E1 and so forth.

Suppose the contour **of** integration, C is taken such that it include only these zeros and no

others, then we can write

n = 1

�

χodz +

¯h C

1

� �

−¯h χ2dz + . . . (3.23)

2πi c C

59

Each **of** these terms involves E − V in the denominator. At the classical turning points where

V (z) = E, we have poles and we can use the residue theorem to evaluate the integrals. For

example, χ1 has a pole at each turnining point with residue −1/4 at each point, hence,

1

2πi

The next term we evaluate by integration by parts

Hence, we can write

Putting it all together

�

C

�

V ′′

�

dz = −3

(E − V (z)) 3/2 2 C

C

χ2(z)dz =

n + 1/2 = 1

h

−

�

C

χ1dz = − 1

. (3.24)

2

1

32(2m) 1/2

�

C

�

c

�

2m(E − V (z))dz

h

128π 2 (2m) 1/2

�

c

V ′2

dz. (3.25)

(E − V (z)) 5/2

V ′2

dz. (3.26)

(E − V (z)) 5/2

V ′2

dz + . . . (3.27)

(E − V (z)) 5/2

Granted, the above analysis is pretty formal! But, what we have is something new. Notice that

we have an extra 1/2 added here that we did not have in the original Bohr-Sommerfield (BS)

theory. What we have is something even more general. The original BS idea came from the

notion that energies and frequencies were related by integer multiples **of** h. But this is really

only valid for transitions between states. If we go back and ask what happens at n = 0 in

the Bohr-Sommerfield theory, this corresponds to a phase-space ellipse with major and minor

axes both **of** length 0–which violates the Heisenberg Uncertainly rule. This new quantization

condition forces the system to have some lowest energy state with phase-space area 1/2.

Where did this extra 1/2 come from? It originates from the classical turning points where

V (x) = E. Recall that for a 1D system bound by a potential, there are at least two such points.

Each contributes a π/4 to the phase. We will see this more explicitly in the next section.

3.2.2 WKB Wavefunction

Going back to our original wavefunction in Eq. 3.14 and writing

ψ = e iS/¯h

where S is the integral **of** χ, we can derive equations for S:

� �

1 ∂S

−

2m ∂x

i¯h

2m

∂2S + V (x) = E. (3.28)

∂x2 Again, as above, one can seek a series expansion **of** S in powers **of** ¯h. The result is simply the

integral **of** Eq. 3.17.

S = So + ¯h

i S1 + . . . (3.29)

60

If we make the approximation that ¯h = 0 we have the classical Hamilton-Jacobi equation for the

action, S. This, along with the definition **of** the momentum, p = dSo/dx = χo, allows us to make

a very firm contact between quantum mechanics and the motion **of** a classical particle.

Looking at Eq. 3.28, it is clear that the classical approximation is valid when the second term

is very small compared to the first. i.e.

¯h d

dx

� dS

dx

¯h S′′

S ′2

� � �2 dx

dS

≪

≪

1

1

¯h d 1

dx p

≪ 1 (3.30)

where we equate dS/dx = p. Since p is related to the de Broglie wavelength **of** the particle

λ = h/p , the same condition implies that

� �

�

� 1 dλ�

�

� � ≪ 1. (3.31)

�2π

dx �

Thus the semi-classical approximation is only valid when the wavelength **of** the particle as determined

by λ(x) = h/p(x) varies slightly over distances on the order **of** the wavelength itself.

Written another way by noting that the gradiant **of** the momentum is

dp

dx

d �

=

dx

Thus, we can write the classical condition as

2m(E − V (x)) = − m

p

dV

dx .

m¯h|F |/p 3 ≪ 1 (3.32)

Consequently, the semi-classical approximation can only be used in regions where the momentum

is not too small. This is especially important near the classical turning points where p → 0. In

classical mechanics, the particle rolls to a stop at the top **of** the potential hill. When this happens

the de Broglie wavelength heads **of**f to infinity and is certainly not small!

Exercise 3.3 Verify the force condition given by Eq. 3.32.

Going back to the expansion for χ

or equivalently for S1

So,

χ1 = − 1

2

χ ′ o

χo

= 1 V

4

′

E − V

S ′ 1 = − S′′ o p′

= − ′

2S

2p

S1(x) = − 1

log p(x)

2

61

(3.33)

(3.34)

If we stick to regions where the semi-classical condition is met, then the wavefunction becomes

ψ(x) ≈ C1

� �

i

p(x)dx C2 i

p(x)dx

� e ¯h + � e− ¯h

p(x) p(x)

(3.35)

The 1/ √ p prefactor has a remarkably simple interpretation. The probability **of** find the particle

in some region between x and x+dx is given by |ψ| 2 so that the classical probability is essentially

proportional to 1/p. So, the fast the particle is moving, the less likely it is to be found in some

small region **of** space. Conversly, the slower a particle moves, the more likely it is to be found in

that region. So the time spend in a small dx is inversely proportional to the momentum **of** the

particle. We will return to this concept in a bit when we consider the idea **of** time in quantum

mechanics.

The C1 and C2 coefficients are yet to be determined. If we take x = a to be one classical

turning point so that x > a corresponds to the classically inaccessible region where E < V (x),

then the wavefunction in that region must be exponentially damped:

ψ(x) ≈ C

�

|p| exp

�

− 1

� x �

|p(x)|dx

¯h a

To the left **of** x = a, we have a combination **of** incoming and reflected components:

ψ(x) = C1

� �

i a �

√p exp pdx +

¯h x

C2

�

√p exp − i

� a �

pdx

¯h x

3.2.3 Semi-classical Tunneling and Barrier Penetration

(3.36)

(3.37)

Before solving the general problem **of** how to use this in an arbitrary well, let’s consider the case

for tunneling through a potential barrier that has some bumpy top or corresponds to some simple

potential. So, to the left **of** the barrier the wavefunction has incoming and reflected components:

Inside we have

ψB(x) =

and to the right **of** the barrier:

ψL(x) = Ae ikx + Be −ikx . (3.38)

C

�

|p(x)|

�

i

|p|dx

e+ ¯h +

D

�

|p(x)|

�

i

|p|dx

e− ¯h

(3.39)

ψR(x) = F e +ikx . (3.40)

If F is the transmitted amplitude, then the tunneling probability is the ratio **of** the transmitted

probability to the incident probability: T = |F | 2 /|A| 2 . If we assume that the barrier is high or

broad, then C = 0 and we obtain the semi-classical estimate for the tunneling probability:

T ≈ exp

�

− 2

� �

b

|p(x)|dx

¯h a

62

(3.41)

where a and b are the turning points on either side **of** the barrier.

Mathematically, we can “flip the potential upside down” and work in imaginary time. In this

case the action integral becomes

S =

� b

a

�

2m(V (x) − E)dx. (3.42)

So we can think **of** tunneling as motion under the barrier in imaginary time.

There are a number **of** useful applications **of** this formula. Gamow’s theory **of** alpha-decay is

a common example. Another useful application is in the theory **of** reaction rates where we want

to determine tunneling corrections to the rate constant for a particular reaction. Close to the top

**of** the barrier, where tunneling may be important, we can expand the potential and approximate

the peak as an upside down parabola

V (x) ≈ Vo − k

2 x2

where +x represents the product side and −x represents the reactant side. See Fig. 3.1 Set the

zero in energy to be the barrier height, Vo so that any transmission for E < 0 corresponds to

tunneling. 1

0

-0.2

-0.4

-0.6

-0.8

e

-4 -2 2 4 x

Figure 3.1: Eckart Barrier and parabolic approximation **of** the transition state

At sufficiently large distances from the turning point, the motion is purely quasi-classical and

we can write the momentum as

�

p = 2m(E + kx2 /2) ≈ x √ �

mk + E m/k/x (3.43)

and the asymptotic for **of** the Schrödinger wavefunction is

ψ = Ae +iξ2 /2 ξ +iɛ−1/2 + Be −iξ 2 /2 ξ −iɛ−1/2

(3.44)

where A and B are coefficients we need to determine by the matching condition and ξ and ɛ are

dimensionless lengths and energies given by ξ = x(mk/¯h) 1/4 �

, and ɛ = (E/¯h) m/k.

1 The analysis is from Kembel, 1935 as discussed in Landau and Lifshitz, QM

63

The particular case we are interested in is for a particle coming from the left and passing to

the right with the barrier in between. So, the wavefunctions in each **of** these regions must be

and

ψR = Be +iξ2 /2 ξ iɛ−1/2

ψL = e −iξ2 /2 (−ξ) −iɛ−1/2 + Ae +iξ 2 /2 (−ξ) iɛ−1/2

(3.45)

(3.46)

where the first term is the incident wave and the second term is the reflected component. So,

|A| 2 | is the reflection coefficient and |B| 2 is the transmission coefficient normalized so that

|A| 2 + |B| 2 = 1.

Lets move to the complex plane and write a new coordinate, ξ = ρe iφ and consider what happens

as we rotate around in φ and take ρ to be large. Since iξ 2 = ρ 2 (i cos 2φ − sin 2φ), we have

and at φ = π

ψR(φ = 0) = Be iρ2

ρ +iɛ−1/2

ψL(φ = 0) = Ae iρ2

(−ρ) +iɛ−1/2

ψR(φ = π) = Be iρ2

(−ρ) +iɛ−1/2

ψL(φ = π) = Ae iρ2

ρ +iɛ−1/2

So, in otherwords, ψR(φ = π) looks like ψL(φ = 0) when

A = B(e iπ ) iɛ−1/2

(3.47)

(3.48)

So, we have the relation A = −iBe−πɛ . Finally, after we normalize this we get the transmission

coefficient:

T = |B| 2 1

=

1 + e−2πɛ which must hold for any energy. If the energy is large and negative, then

T ≈ e −2πɛ .

Also, we can compute the reflection coefficient for E > 0 as 1 − D,

R =

1

.

1 + e +2πɛ

Exercise 3.4 Verify these last relationships by taking the ψR and ψL, performing the analytic

continuation.

64

This gives us the transmission probabilty as a function **of** incident energy. But, normal

chemical reactions are not done at constant energy, they are done at constant temperature.

To get the thermal transmission coefficient, we need to take a Boltzmann weighted average **of**

transmission coefficients

Tth(β) = 1

�

Z

dEe −Eβ T (E) (3.49)

where β = 1/kT and Z is the partition function. If E represents a continuum **of** energy states

then

Tth(β) = − βω¯h(ψ(0) ( βω¯h

4π ) − ψ(0) ( 1 βω¯h

( 4 π

4π

+ 2)))

(3.50)

where ψ (n) (z) is the Polygamma function which is the nth derivative **of** the digamma function,

ψ (0) (z), which is the logarithmic derivative **of** Eulers gamma function, ψ (0) (z) = Γ(z)/Γ(z). 2

3.3 Connection Formulas

In what we have considered thus far, we have assumed that up until the turning point the

wavefunction was well behaved and smooth. We can think **of** the problem as having two domains:

an exterior and an interior. The exterior part we assumed to be simple and the boundary

conditions trivial to impose. The next task is to figure out the matching condition at the turning

point for an arbitrary system. So far what we have are two pieces, ψL and ψR, in the notation

above. What we need is a patch. To do so, we make a linearizing assumption for the force at

the classical turning point:

E − V (x) ≈ Fo(x − a) (3.51)

where Fo = −dV/dx evaluated at x = a. Thus, the phase integral is easy:

�

1 x

pdx =

¯h a

2 �

2mFo(x − a)

3¯h

3/2

(3.52)

But, we can do better than that. We can actually solve the Schrodinger equation for the linear

potential and use the linearized solutions as our patch. The Mathematica Notebook AiryFunctions.nb

goes through the solution **of** the linearized Schrodinger equation

which can be re-written as

with

2 See the Mathematica Book, sec 3.2.10.

− ¯h2 dψ

2m

dx 2 + (E + V ′ )ψ = 0 (3.53)

α =

ψ ′′ = α 3 xψ (3.54)

�

2m

¯h 2 V ′ �1/3

(0) .

65

Absorbing the coefficient into a new variable y, we get Airy’s equation

ψ ′′ (y) = yψ.

The solutions **of** Airy’s equation are Airy functions, Ai(y) and Bi(y) for the regular and irregular

cases. The integral representation **of** the Ai and Bi are

and

Bi(y) = 1

π

Plots **of** these functions are shown in Fig. 3.2.

Ai@yD, Bi@yD

1.5

-10 -8 -6 -4 -2 2

1

0.5

-0.5

Ai(y) = 1

� � �

∞ 3 s

cos + sy ds (3.55)

π 0 3

� �

∞

e

0

−s3 � ��

3

/3+sy s

+ sin + sy ds (3.56)

3

y

Figure 3.2: Airy functions, Ai(y) (red) and Bi(y) (blue)

Since both Ai and Bi are acceptible solutions, we will take a linear combination **of** the two

as our patching function and figure out the coefficients later.

ψP = aAi(αx) + bBi(αx) (3.57)

We now have to determine those coefficients. We need to make two assumptions. One,

that the overlap zones are sufficiently close to the turning point that a linearized potential is

reasonable. Second, the overlap zone is far enough from the turning point (at the origin) that

the WKB approximation is accurate and reliable. You can certainly cook up some potential

for which this will not work, but we will assume it’s reasonable. In the linearized region, the

momentum is

So for +x,

� x

0

p(x) = ¯hα 3/2 (−x) 3/2

(3.58)

|p(x)|dx = 2¯h(αx) 3/2 /3 (3.59)

66

and the WKB wavefunction becomes:

ψR(x) =

D

√ ¯hα 3/4 x 1/4 e−2(αx)3/2 /3 . (3.60)

In order to extend into this region, we will use the asymptotic form **of** the Ai and Bi functions

for y ≫ 0

Ai(y) ≈ e−2y3/2 /3

2 √ πy 1/4

Clearly, the Bi(y) term will not contribute, so b = 0 and

�

4π

a =

α¯h D.

(3.61)

Bi(y) ≈ e+2y3/2 /3

√ . (3.62)

πy1/4 Now, for the other side, we do the same proceedure. Except this time x < 0 so the phase

integral is

� 0

pdx = 2¯h(−αx) 3/2 /3. (3.63)

Thus the WKB wavefunction on the left hand side is

ψL(x) = 1

√ p

=

x

�

Be 2i(−αx)3/2 /3 + Ce −2i(−αx) 3/2 /3 �

1 �

√ Be

¯hα3/4 (−x) 1/4

2i(−αx)3/2 /3 −2i(−αx)

+ Ce 3/2 /3 �

(3.64)

(3.65)

That’s the WKB part, to connect with the patching part, we again use the asymptotic forms for

y ≪ 0 and take only the regular solution,

Ai(y) ≈

≈

1 �

√ sin 2(−y)

π(−y) 1/4 3/2 /3 + π/4 �

1

2i √ π(−y) 1/4

�

e iπ/4 e i2(−y)3/2 /3 −iπ/4 −i2(−y)

− e e 3/2 /3 �

Comparing the WKB wave and the patching wave, we can match term-by-term

a

2i √ π eiπ/4 = B

√

¯hα

−a

2i √ π e−iπ/4 = C

√

¯hα

(3.66)

(3.67)

(3.68)

Since we know a in terms **of** the normalization constant D, B = ie iπ/4 D and C = ie −iπ/4 . This

is the connection! We can write the WKB function across the turning point as

⎧

⎪⎨

ψW KB(x) =

⎪⎩

√2D sin

p(x) � 1

1

√ 2D −

e ¯h

|p(x)|

� 0

¯h x

� 0

x pdx

67

pdx + π/4�

x < 0

x > 0

(3.69)

node xn

1 -2.33811

2 -4.08795

3 -5.52056

4 -6.78671

5 -7.94413

6 -9.02265

7 -10.0402

Table 3.1: Location **of** nodes for Airy, Ai(x) function.

Example: Bound states in the linear potential

Since we worked so hard, we have to use the results. So, consider a model problem for a particle

in a gravitational field. Actually, this problem is not so far fetched since one can prepare trapped

atoms above a parabolic reflector and make a quantum bouncing ball. Here the potential is

V (x) = mgx where m is the particle mass and g is the graviational constant (g = 9.80m/s).

We’ll take the case where the reflector is infinite so that the particle cannot penetrate into it.

The Schrödinger equation for this potential is

− ¯h2

2m ψ′′ + (E − mgx)ψ = 0. (3.70)

The solutions are the Airy Ai(x) functions. Setting, β = mg and c = ¯h 2 /2m, the solutions are

ψ = CAi(

�

− β

c

� 1/3

(x − E/β)) (3.71)

However, there is one caveat: ψ(0) = 0, thus the Airy functions must have their nodes at x = 0.

So we have to systematically shift the Ai(x) function in x until a node lines up at x = 0. The

nodes **of** the Ai(x) function can be determined and the first 7 **of** them are To find the energy

levels, we systematically solve the equation

�

− β

�1/3 En

c β

= xn

So the ground state is where the first node lands at x = 0,

E1 = 2.33811β

(β/c) 1/3

= 2.33811mg

(2m 2 g/¯h 2 ) 1/3

(3.72)

and so on. Of course, we still have to normalize the wavefunction to get the correct energy.

We can make life a bit easier by using the quantization condition derived from the WKB

approximation. Since we require the wavefunction to vanish exactly at x = 0, we have:

�

1 xt

p(x)dx +

¯h 0

π

= nπ. (3.73)

4

68

15

10

5

-5

-10

2 4 6 8 10

Figure 3.3: Bound states in a graviational well

This insures us that the wave vanishes at x = 0, xt in this case is the turning point E = mgxt.

(See Figure 3.3) As a consequence,

Since p(x) =

� xt

0

� xt

0

p(x)dx = (n − 1/4)π

�

2m(En − mgx), The integral can be evaluated

�

2m(E − mghdx = √ ⎛

2 ⎝ 2En

√ �

⎞

Enm 2 m (En − gmxt) (−En + gmxt)

+ ⎠ (3.74)

3gm

3gm

Since xt = En/mg for the classical turning point, the phase intergral becomes

{ 2√2En 2

3g √ } = (n − 1/4)π¯h.

Enm

Solving for En yields the semi-classical approximation for the eigenvalues:

En =

2

g 3 m 1 �

1

2

2� 3

3 (1 − 4 n) (3 π) 3 ¯h 2

3

4 2 1

3

(3.75)

In atomic units, the gravitional constant is g = 1.08563 × 10 −22 bohr/au 2 (Can you guess why

we rarely talk about gravitational effects on molecules?). For n = 0, we get for an electron

E sc

o = 2.014 × 10 −15 hartree or about 12.6 Hz. So, graviational effects on electrons are extremely

tiny compared to the electron’s total energy.

69

3.4 Scattering

b

r

θ

Figure 3.4: Elastic scattering trajectory for classical collision

The collision between two particles plays an important role in the dynamics **of** reactive molecules.

We consider here the collosion between two particles interacting via a central force, V (r). Working

in the center **of** mass frame, we consider the motion **of** a point particle with mass µ with

position vector �r. We will first examine the process in a purely classical context since it is

intuitive and then apply what we know to the quantum and semiclassical case.

3.4.1 Classical Scattering

The angular momentum **of** the particle about the origin is given by

θ c

r c

�L = �r × �p = µ(�r × ˙ �r) (3.76)

we know that angular momentum is a conserved quantity and it is is easy to show that ˙ � L = 0

viz.

˙�L = d

dt �r × �p = (˙ �r × ˙ �r + (�r × ˙ �p). (3.77)

Since ˙r = ˙p/µ, the first term vanishes; likewise, the force vector, ˙ �p = −dV/dr, is along �r so that

the second term vanishes. Thus, L = const meaning that angular momentum is a conserved

quantity during the course **of** the collision.

In cartesian coordinates, the total energy **of** the collision is given by

To convert from cartesian to polar coordinates, we use

E = µ

2 ( ˙x2 + ˙y 2 ) + V. (3.78)

x = r cos θ (3.79)

70

χ

Thus,

where we use the fact that

y = r sin θ (3.80)

˙x = ˙r cos θ − r ˙ θ sin θ (3.81)

˙y = ˙r sin θ + r ˙ θ cos θ. (3.82)

E = mu

2 ˙r2 + V (r) + L2

2µr 2

L = µr 2 ˙ θ 2

(3.83)

(3.84)

where L is the angular momentum. What we see here is that we have two potential contributions.

The first is the physical attraction (or repulsion) between the two scattering bodies. The second is

a purely repulsive centrifugal potential which depends upon the angular momentum and ultimatly

upon the inpact parameters. For cases **of** large impact parameters, this can be the dominant

force. The effective radial force is given by

µ¨r = L2

2r3 ∂V

−

µ ∂r

(3.85)

Again, we note that the centrifugal contribution is always repulsive while the physical interaction

V(r) is typically attractive at long range and repulsive at short ranges.

We can derive the solutions to the scattering motion by integrating the velocity equations for

r and θ

˙r =

� �

2

± E − V (r) −

µ

L2

2µr2 ��1/2 ˙θ = L

µr2 (3.86)

(3.87)

and taking into account the starting conditions for r and θ. In general, we could solve the

equations numerically and obtain the complete scatering path. However, really what we are

interested in is the deflection angle χ since this is what is ultimately observed. So, we integrate

the last two equations and derive θ in terms **of** r.

θ(r) =

� θ

= −

0

� r

� r

dθ = −

∞

L

µr 2

∞

dθ

dr (3.88)

dr

1

�

(2/µ)(E − V − L 2 /2µr 2 )

dr (3.89)

where the collision starts at t = −∞ with r = ∞ and θ = 0. What we want to do is derive this

in terms **of** an impact parameter, b, and scattering angle χ. These are illustrated in Fig. 3.4 and

can be derived from basic kinematic considerations. First, energy is conserved through out, so if

71

we know the asymptotic velocity, v, then E = µv 2 /2. Secondly, angular momentum is conserved,

so L = µ|r × v| = µvb. Thus the integral above becomes

� r

θ(r) = −b

= −

∞

� r

∞

dθ

dr (3.90)

dr

dr

r2 �

1 − V/E − b2 . (3.91)

/r2 Finally, the angle **of** deflection is related to the angle **of** closest approach by 2θc + χ = π; hence,

� ∞

χ = π − 2b

rc

dr

r2 �

1 − V/E − b2 /r2 The radial distance **of** closest approach is determined by

which can be restated as

E = L2

2µr 2 c

b 2 = r 2 c

�

1 −

(3.92)

+ V (rc) (3.93)

�

V (rc)

E

(3.94)

Once we have specified the potential, we can compute the deflection angle using Eq.3.94. If

V (rc) < 0 , then rc < b and we have an attractive potential, if V (rc) > 0 , then rc > b and the

potential is repulsive at the turning point.

If we have a beam **of** particles incident on some scattering center, then collisions will occur

with all possible impact parameter (hence angular momenta) and will give rise to a distribution

in the scattering angles. We can describe this by a differential cross-section. If we have some

incident intensity **of** particles in our beam, Io, which is the incident flux or the number **of** particles

passing a unit area normal to the beam direction per unit time, then the differential cross-section,

I(χ), is defined so that I(χ)dΩ is the number **of** particles per unit time scattered into some solid

angle dΩ divided by the incident flux.

The deflection pattern will be axially symmetric about the incident beam direction due the

spherical symmetry **of** the interaction potential; thus, I(χ) only depends upon the scattering angle.

Thus, dΩ can be constructed by the cones defining χ and χ+dχ, i.e. dΩ = 2π sin χdχ. Even

if the interaction potential is not spherically symmetric, since most molecules are not spherical,

the scattering would be axially symmetric since we would be scattering from a homogeneous distribution

**of** al possible orientations **of** the colliding molecules. Hence any azimuthal dependency

must vanish unless we can orient or the colliding species.

Given an initial velocity v, the fraction **of** the incoming flux with impact parameter between

b and b + db is 2πbdb. These particles will be deflected between χ and χ + dχ if dχ/db > 0 or

between χ and χ − dχ if dχ/db < 0. Thus, I(χ)dΩ = 2πbdb and it follows then that

I(χ) =

b

. (3.95)

sin χ|dχ/db|

72

Thus, once we know χ(b) for a given v, we can get the differential cross-section. The total

cross-section is obtained by integrating

σ = 2π

� π

0

I(χ) sin χdχ. (3.96)

This is a measure **of** the attenuation **of** the incident beam by the scattering target and has the

units **of** area.

3.4.2 Scattering at small deflection angles

Our calculations will be greatly simplified if we consider collisions that result in small deflections

in the forward direction. If we let the initial beam be along the x axis with momentum p, then

the scattered momentum, p ′ will be related to the scattered angle by p ′ sin χ = p ′ y. Taking χ to

be small

χ ≈ p′ y momentum transfer

= . (3.97)

p ′ momentum

Since the time derivative **of** momentum is the force, the momentum transfered perpendicular to

the incident beam is obtained by integrating the perpendicular force

F ′ y = − ∂V

∂y

= −∂V

∂r

∂r

∂y

where we used r 2 = x 2 + y 2 and y ≈ b. Thus we find,

χ =

p ′ y

µ(2E/µ) 1/2

= −b(2µE) −1/2

= −b(2µE) −1/2

� +∞

−∞

� 2E

µ

= −∂V

∂r

∂V dt

∂r r

b

r

� −1/2 � +∞

−∞

∂V

∂r

dx

r

(3.98)

(3.99)

(3.100)

(3.101)

= − b

� ∞ ∂V

E b ∂r (r2 − b 2 ) −1/2 dr (3.102)

where we used x = (2E/µ) 1/2t and x varies from −∞ to +∞ as r goes from −∞ to b and back.

Let us use this in a simple example **of** the V = C/rs potential for s > 0. Substituting V into

the integral above and solving yields

χ = sCπ1/2

2b s E

Γ((s + 1)/2)

. (3.103)

Γ(s/2 + 1)

This indicates that χE ∝ b−s and |dχ/db| = χs/b.

differential cross-section

Thus, we can conclude by deriving the

I(χ) = 1

s χ−(2+2/s)

� �

1/2

2/s

sCπ Γ((s + 1)/2)

2E Γ(s/2 + 1)

(3.104)

for small values **of** the scattering angle. Consequently, a log-log plot **of** the center **of** mass

differential cross-section as a function **of** the scattering angle at fixed energy should give a straight

line with a slope −(2 + 2/s) from which one can determine the value **of** s. For the van der Waals

potential, s = 6 and I(χ) ∝ E −1/3 χ −7/3 .

73

3.4.3 **Quantum** treatment

The quantum mechanical case is a bit more complex. Here we will develop a brief overview

**of** quantum scattering and move onto the semiclassical evaluation. The quantum scattering is

determined by the asymptotic form **of** the wavefunction,

ψ(r, χ) r→∞

�

−→ A e ikz + f(χ)

r eikr

�

(3.105)

where A is some normalization constant and k = 1/λ = µv/¯h is the initial wave vector along

the incident beam direction (χ = 0). The first term represents a plane wave incident upon

the scatterer and the second represents an out going spherical wave. Notice that the outgoing

amplitude is reduced as r increases. This is because the wavefunction spreads as r increases. If

we can collimate the incoming and out-going components, then the scattering amplitude f(χ) is

related to the differential cross-section by

I(χ) = |f(χ)| 2 . (3.106)

What we have is then that asymptotic form **of** the wavefunction carries within it information

about the scattering process. As a result, we do not need to solve the wave equation for all **of**

space, we just need to be able to connect the scattering amplitude to the interaction potential.

We do so by expanding the wave as a superposition **of** Legendre polynomials

∞�

ψ(r, χ) = Rl(r)Pl(cos χ). (3.107)

l=0

Rl(r) must remain finite as r = 0. This determines the form **of** the solution.

When V (r) = 0, then ψ = A exp(ikz) and we can expand the exponential in terms **of** spherical

waves

e ikz ∞�

ilπ/2 sin(kr − lπ/2)

= (2l + 1)e Pl(cos χ) (3.108)

l=0

kr

= 1

∞�

(2l + 1)i

2i l=0

l �

i(kr−lπ/2) e

Pl(cos χ)

+

kr

e−i(kr−lπ/2)

�

(3.109)

kr

We can interpret this equation in the following intuitive way: the incident plane wave is equivalent

to an infinite superposition **of** incoming and outgoing spherical waves in which each term

corresponds to a particular angular momentum state with

�

L = ¯h l(l + 1) ≈ ¯h(l + 1/2). (3.110)

From our analysis above, we can relate L to the impact parameter, b,

b = L

µv

l + 1/2

≈ λ. (3.111)

k

In essence the incoming beam is divided into cylindrical zones in which the lth zone contains

particles with impact parameters (and hence angular momenta) between lλ and (l + 1)λ.

74

Exercise 3.5 The impact parameter, b, is treated as continuous; however, in quantum mechanics

we allow only discrete values **of** the angular momentum, l. How will this affect our results, since

b = (l + 1/2)λ from above.

If V (r) is short ranged (i.e. it falls **of**f more rapidly than 1/r for large r), we can derive a

general solution for the asymptotic form

∞�

� � ��

lπ sin(kr − lπ/2 + ηl

ψ(r, χ) −→ (2l + 1) exp i + ηl

Pl(cos χ). (3.112)

2 kr

l=0

The significant difference between this equation and the one above for the V (r) = 0 case is the

addition **of** a phase-shift ηl. This shift only occurs in the outgoing part **of** the wavefunction and so

we conclude that the primary effect **of** a potential in quantum scattering is to introduce a phase

in the asymptotic form **of** the scattering wave. This phase must be a real number and has the

following physical interpretation illustrated in Fig. 3.5 A repulsive potential will cause a decrease

in the relative velocity **of** the particles at small r resulting in a longer de Broglie wavelength. This

causes the wave to be “pushed out” relative to that for V = 0 and the phase shift is negative.

An attractive potential produces a positive phase shift and “pulls” the wavefunction in a bit.

Furthermore, the centrifugal part produces a negative shift **of** −lπ/2.

Comparing the various forms for the asymptotic waves, we can deduce that the scattering

amplitude is given by

f(χ) = 1

2ik

From this, the differential cross-section is

I(χ) = λ 2

�

� ∞� �

� (2l + 1)e

�

iηl

�

�

�

sin(ηl)Pl(cos χ) �

�

∞�

(2l + 1)(e

l=0

2iηl − 1)Pl(cos χ). (3.113)

l=0

2

(3.114)

What we see here is the possibility for interference between different angular momentum components

Moving forward at this point requires some rather sophisticated treatments which we reserve

for a later course. However, we can use the semiclassical methods developed in this chapter to

estimate the phase shifts.

3.4.4 Semiclassical evaluation **of** phase shifts

The exact scattering wave is not so important. What is important is the asymptotic extent **of**

the wavefunction since that is the part which carries the information from the scattering center

to the detector. What we want is a measure **of** the shift in phase between a scattering with and

without the potential. From the WKB treatment above, we know that the phase is related to

the classical action along a given path. Thus, in computing the semiclassical phase shifts, we

are really looking at the difference between the classical actions for a system with the potential

switched on and a system with the potential switched **of**f.

η SC

l

= lim

R→∞

� � R

rc

dr

λ(r) −

� R dr

b λ(r)

75

�

(3.115)

1

0.5

-0.5

-1

5 10 15 20 25 30

Figure 3.5: Form **of** the radial wave for repulsive (short dashed) and attractive (long dashed)

potentials. The form for V = 0 is the solid curve for comparison.

R is the radius a sphere about the scattering center and λ(r) is a de Broglie wavelength

λ(r) = ¯h

p

= 1

k(r) =

¯h

µv(1 − V (r) − b 2 /r 2 ) 1/2

(3.116)

associated with the radial motion. Putting this together:

η SC

l = lim

R→∞ k

⎛

� R V (r) b2

⎝ (1 − −

rc E r2 )1/2 � �

R

dr − 1 −

b

b2

r2 =

� ⎞

1/2

dr⎠

⎛

� R

� �

R

lim ⎝ k(r)dr − k 1 −

R→∞ rc

b

(3.117)

b2

r2 � ⎞

1/2

dr⎠

(3.118)

(k is the incoming wave-vector.) The last integral we can evaluate:

� R

k

b

(r 2 − b 2 )) 1/2

r

dr = k

�

(r 2 − b 2 −1 b

) − b cos

r

��

���� R

b

= kR − kbπ

2

(3.119)

Now, to clean things up a bit, we add and subtract an integral over k. (We do this to get rid **of**

the R dependence which will cause problems when we take the limit R → ∞.)

η SC

�� R

l = lim

R→∞ rc

� R

=

=

rc

� R

rc

k(r)dr −

� R

rc

kdr +

� R

rc

kdr − (kR − kbπ

2 )

�

(3.120)

(k(r) − k)dr − k(rc − bπ/2) (3.121)

(k(r) − k)dr − krcπ(l + 1/2)/2 (3.122)

This last expression is the standard form **of** the phase shift.

The deflection angle can be determined in a similar way.

�� �

� � �

��

χ = lim

R→∞

π − 2 dθ

actual path

− π − dθ

V =0 path

76

(3.123)

We transform this into an integral over r

⎡

� �

∞ V (r) b2

χ = −2b ⎣ 1 − −

E r2 �−1/2 � �

dr ∞

− 1 −

r2 b

b2

r2 �−1/2 dr

r2 ⎤

⎦ (3.124)

rc

Agreed, this is weird way to express the scattering angle. But let’s keep pushing this forward.

The last integral can be evaluated

� �

∞

1 −

b

b2

r2 �−1/2 �

dr 1 b �

�

= cos−1 �

r2 b r �

∞

b

= − π

. (3.125)

2b

which yields the classical result we obtained before. So, why did we bother? From this we can

derive a simple and useful connection between the classical deflection angle and the rate **of** change

**of** the semiclassical phase shift with angular momentum, dη SC

l /dl. First, recall the Leibnitz rule

for taking derivatives **of** integrals:

�

d b(x)

f(x, y)dy =

dx a(x)

b

dx

Taking the derivative **of** η SC

l

(∂b/∂l)E = b/k, we find that

�

da

b(x)

f(b(x), y) − f(a(x), y) +

dx a(x)

∂f(x, y)

dy (3.126)

∂x

with respect to l, using the last equation a and the relation that

dη SC

l

dl

χ

= . (3.127)

2

Next, we examine the differential cross-section, I(χ). The scattering amplitude

f(χ) = λ

2i

∞�

(2l + 1)e

l=0

2iηlPl(cos χ). (3.128)

where we use λ = 1/k and exclude the singular point where χ = 0 since this contributes nothing

to the total flux.

Now, we need a mathematical identity to take this to the semiclassical limit where the potential

varies slowly with wavelength. What we do is to first relate the Legendre polynomial,

Pl(cos θ) to a zeroth order Bessel function for small values **of** θ (θ ≪ 1).

Pl(cos θ) = J0((l + 1/2)θ). (3.129)

Now, when x = (l + 1/2)θ ≫ 1 (i.e. large angular momentum), we can use the asymptotic

expansion **of** J0(x)

Pulling this together,

Pl(cos θ) →

�

2

π(l + 1/2)θ

�

2

J0(x) →

πx sin

�

x + π

�

. (3.130)

4

�1/2 �

sin ((l + 1/2)θ + π/4) ≈

77

2

π(l + 1/2)

�1/2 sin ((l + 1/2)θ + π/4)

(sin θ) 1/2 (3.131)

for θ(l + 1/2) ≫ 1. Thus, we can write the semi-classical scattering amplitude as

where

f(χ) = −λ

∞�

l=0

� �1/2 (l + 1/2) �

e

2π sin χ

iφ+

+ e iφ−�

(3.132)

φ ± = 2ηl ± (l + 1/2)χ ± π/4. (3.133)

The phases are rapidly oscillating functions **of** l. Consequently, the majority **of** the terms must

cancel and the sum is determined by the ranges **of** l for which either φ + or φ − is extremized.

This implies that the scattering amplitude is determined almost exclusively by phase-shifts which

satisfy

2 dηl

dl

± χ = 0, (3.134)

where the + is for dφ + /dl = 0 and the − is for dφ − /dl = 0. This demonstrates that the only the

phase-shifts corresponding to impact parameter b can contribute significantly to the differential

cross-section in the semi-classical limit. Thus, the classical condition for scattering at a given

deflection angle, χ is that l be large enough for Eq. 3.134 to apply.

3.4.5 Resonance Scattering

3.5 Problems and Exercises

Exercise 3.6 In this problem we will (again) consider the ammonia inversion problem, this time

we will proceed in a semi-classical context.

Recall that the ammonia inversion potential consists **of** two symmetrical potential wells separated

by a barrier. If the barrier was impenetrable, one would find energy levels corresponding to

motion in one well or the other. Since the barrier is not infinite, there can be passage between

wells via tunneling. This causes the otherwise degenerate energy levels to split.

In this problem, we will make life a bit easier by taking

V (x) = α(x 4 − x 2 )

as in the examples in Chapter 5.

Let ψo be the semi-classical wavefunction describing the motion in one well with energy Eo.

Assume that ψo is exponentially damped on both sides **of** the well and that the wavefunction

is normalized so that the integral over ψ 2 o is unity. When tunning is taken into account, the

wavefunctions corresponding to the new energy levels, E1 and E2 are the symmetric and antisymmetric

combinations **of** ψo(x) and ψo(−x)

ψ1 = (ψo(x) + ψo(−x)/ √ 2

ψ2 = (ψo(x) − ψo(−x)/ √ 2

where ψo(−x) can be thought **of** as the contribution from the zeroth order wavefunction in the

other well. In Well 1, ψo(−x) is very small and in well 2, ψo(+x) is very small and the product

ψo(x)ψo(−x) is vanishingly small everywhere. Also, by construction, ψ1 and ψ2 are normalized.

78

1. Assume that ψo and ψ1 are solutions **of** the Schrödinger equations

and

ψ ′′

o + 2m

¯h 2 (Eo − V )ψo = 0

ψ ′′

1 + 2m

¯h 2 (E1 − V )ψ1 = 0,

Multiply the former by ψ1 and the latter by ψo, combine and subtract equivalent terms, and

integrate over x from 0 to ∞ to show that

Perform a similar analysis to show that

E1 − Eo = − ¯h2

m ψo(0)ψ ′ o(0),

E2 − Eo = + ¯h2

m ψo(0)ψ ′ o(0),

2. Show that the unperturbed semiclassical wavefunction is

and

where vo =

ψo(0) =

� ω

2πvo

�

exp − 1

� a �

|p|dx

¯h 0

ψ ′ o(0) = mvo

¯h ψo(0)

�

2(Eo − V (0))/m and a is the classical turning point at Eo = V (a).

3. Combining your results, show that the tunneling splitting is

∆E = ¯hω

π exp

�

− 1

� +a �

|p|dx .

¯h −a

where the integral is taken between classical turning points on either side **of** the barrier.

4. Assuming that the potential in the barrier is an upside-down parabola

what is the tunneling splitting.

V (x) ≈ Vo − kx 2 /2

5. Now, taking α = 0.1 expand the potential about the barrier and compute determine the

harmonic force constant for the upside-down parabola. Using the equations you derived and

compute the tunneling splitting for a proton in this well. How does this compare with the

calculations presented in Chapter 5.

79

Chapter 4

Postulates **of** **Quantum** **Mechanics**

When I hear the words “Schrödinger’s cat”, I wish I were able to reach for my gun.

Stephen Hawkings.

The dynamics **of** physical processes at a microscopic level is very much beyond the realm **of**

our macroscopic comprehension. In fact, it is difficult to imagine what it is like to move about on

the length and timescales for whcih quantum mechanics is important. However, for molecules,

quantum mechanics is an everyday reality. Thus, in order to understand how molecules move and

behave, we must develop a model **of** that dynamics in terms which we can comprehend. Making

a model means developing a consistent mathematical framework in which the mathematical

operations and constructs mimic the physical processes being studied.

Before moving on to develop the mathematical framework required for quantum mechanics,

let us consider a simple thought experiment. WE could do the experiment, however, we would

have to deal with some additional technical terms, like funding. The experiment I want to

consider goes as follows: Take a machine gun which shoots bullets at a target. It’s not a very

accurate gun, in fact, it sprays bullets randomly in the general direction **of** the target.

The distribution **of** bullets or histogram **of** the amount **of** lead accumulated in the target is

roughly a Gaussian, C exp(−x 2 /a). The probability **of** finding a bullet at x is given by

P (x) = Ce −x2 /a . (4.1)

C here is a normalization factor such that the probability **of** finding a bullet anywhere is 1. i.e.

� ∞

−∞

The probability **of** finding a bullet over a small interval is

� b

a

dxP (x) = 1 (4.2)

dxP (x)〉0. (4.3)

Now suppose we have a bunker with 2 windows between the machine gun and the target such

that the bunker is thick enough that the bullets coming through the windows rattle around a

few times before emerging in random directions. Also, let’s suppose we can “color” the bullets

with some magical (or mundane) means s.t. bullets going through 1 slit are colored “red” and

80

Figure 4.1: Gaussian distribution function

1

0.8

0.6

0.4

0.2

-10 -5 5 10

81

Figure 4.2: Combination **of** two distrubitions.

1

0.8

0.6

0.4

0.2

-10 -5 5 10

bullets going throught the other slit are colored “blue”. Thus the distribution **of** bullets at a

target behind the bunker is now

P12(x) = P1(x) + P2(x) (4.4)

where P1 is the distribution **of** bullets from window 1 (the blue bullets) and P2 the “red” bullets.

Thus, the probability **of** finding a bullet that passed through either 1 or 2 is the sum **of** the

probabilies **of** going through 1 and 2. This is shown in Fig. ??

Now, let’s make an “electron gun” by taking a tungsten filiment heated up so that electrons

boil **of**f and can be accellerated toward a phosphor screen after passing through a metal foil with

a pinhole in the middle We start to see little pin points **of** light flicker on the screen–these are

the individual electron “bullets” crashing into the phosphor.

If we count the number **of** electrons which strike the screen over a period **of** time–just as in

the machine gun experiment, we get a histogram as before. The reason we get a histogram is

slightly different than before. If we make the pin hole smaller, the distribution gets wider. This

is a manifestation **of** the Heisenberg Uncertainty Principle which states:

∆x · δp ≥ ¯h/2 (4.5)

In otherwords, the more I restrict where the electron can be (via the pin hole) the more uncertain

I am about which direction is is going (i.e. its momentum parallel to the foil.) Thus, I wind up

with a distribution **of** momenta leaving the foil.

Now, let’s poke another hole in the foil and consider the distribution **of** electrons on the foil.

Based upon our experience with bullets, we would expect:

P12 = P1 + P2

BUT electrons obey quantum mechanics! And in quantum mechanics we represent a particle

via an amplitude. And one **of** the rules **of** quantum mechanics is that we first add amplitudes

and that probabilities are akin to the intensity **of** the combinded amplitudes. I.e.

P = |ψ1 + ψ2| 2

82

(4.6)

(4.7)

Figure 4.3: Constructive and destructive interference from electron/two-slit experiment. The

superimposed red and blue curves are P1 and P2 from the classical probabilities

1

0.8

0.6

0.4

0.2

-10 -5 5 10

where ψ1 and ψ2 are the complex amplitudes associated with going through hole 1 and hole 2.

Since they are complex numbers,

Thus,

ψ1 = a1 + ib1 = |psi1|e iφ1 (4.8)

ψ2 = a2 + ib2 = |psi2|e iφ2 (4.9)

ψ1 + ψ2 = |ψ1|e iφ1 + |ψ2|e iφ2 (4.10)

|ψ1 + ψ2| 2 = (|ψ1|e iφ1 + |ψ2|e iφ2 )

× (|ψ1|e −iφ1 + |ψ2|e −iφ2 ) (4.11)

P12 = |ψ1| 2 + |ψ2| 2 + 2|ψ1||ψ2| cos(φ1 − φ2)

�

(4.12)

P12 = P1 + P2 + 2

P1P2 cos(φ1 − φ2) (4.13)

In other words, I get the same envelope as before, but it’s modulated by the cos(φ1 − φ2)

“interference” term. This is shown in Fig. 4.3. Here the actual experimental data is shown as a

dashed line and the red and blue curves are the P1 and P2. Just as if a wave **of** electrons struck

the two slits and diffracted (or interfered) with itself. However, we know that electrons come in

definite chunks–we can observe individual specks on the screen–only whole lumps arrive. There

are no fractional electrons.

Conjecture 1 Electrons–being indivisible chunks **of** matter–either go through slit 1 or slit 2.

Assuming Preposition 1 is true, we can divide the electrons into two classes:

1. Those that go through slit 1

83

2. Those that go through slit 2.

We can check this preposition by plugging up hole 1 and we get P2 as the resulting distribution.

Plugging up hole 2, we get P1. Perhaps our preposition is wrong and electrons can be split in

half and half **of** it went through slit 1 and half through slit 2. NO! Perhaps, the electron went

through 1 wound about and went through 2 and through some round-about way made its way

to the screen.

Notice that in the center region **of** P12, P12 > P1+P2, as if closing 1 hole actually decreased the

number **of** electrons going through the other hole. It seems very hard to justify both observations

by proposing that the electrons travel in complicated pathways.

In fact, it is very mysterious. And the more you study quantum mechanics, the more mysterious

it seems. Many ideas have been cooked up which try to get the P12 curve in terms **of**

electrons going in complicated paths–all have failed.

Surprisingly, the math is simple (in this case). It’s just adding complex valued amplitudes.

So we conclude the following:

Electrons always arrive in discrete, indivisible chunks–like particles. However, the

probability **of** finding a chunk at a given position is like the distribution **of** the intensity

**of** a wave.

We could conclude that our conjecture is false since P12 �= P1 + P2. This we can test.

Let’s put a laser behind the slits so that an electron going through either slit scatters a bit

**of** light which we can detect. So, we can see flashes **of** light from electrons going through slit

1, flashes **of** light from electrons going through slit 2, but NEVER two flashes at the same

time. Conjecture 1 is true. But if we look at the resulting distribution: we get P12 = P1 + P2.

Measuring which slit the electon passes through destroys the phase information. When we make

a measurement in quantum mechanics, we really disturb the system. There is always the same

amount **of** disturbance because electrons and photons always interact in the same way every time

and produce the same sized effects. These effects “rescatter” the electrons and the phase info is

smeared out.

It is totally impossible to devise an experiment to measure any quantum phenomina without

disturbing the system you’re trying to measure. This is one **of** the most fundimental and perhaps

most disturbing aspects **of** quantum mechanics.

So, once we have accepted the idea that matter comes in discrete bits but that its behavour

is much like that **of** waves, we have to adjust our way **of** thinking about matter and dynamics

away from the classical concepts we are used to dealing with in our ordinary life.

These are the basic building blocks **of** quantum mechanics. Needless to say they are stated in

a rather formal language. However, each postulate has a specific physical reason for its existance.

For any physical theory, we need to be able to say what the system is, how does it move, and

what are the possible outcomes **of** a measurement. These postulates provide a sufficient basis for

the development **of** a consistent theory **of** quantum mechanics.

84

4.0.1 The description **of** a physical state:

The state **of** a physical system at time t is defined by specifying a vector |ψ(t)〉 belonging to a

state space H. We shall assume that this state vector can be normalized to one:

〈ψ|ψ〉 = 1

4.0.2 Description **of** Physical Quantities:

Every measurable physical quantity, A, is described by an operator acting in H; this operator is

an observable.

A consequence **of** this is that any operator related to a physical observable must be Hermitian.

This we can prove. Hermitian means that

〈x|O|y〉 = 〈y|O|x〉 ∗

Thus, if O is a Hermitian operator and 〈O〉 = 〈ψ|O|ψ〉 = λ〈ψ|ψ〉,

Likewise,

If O is Hermitian, we can also write

(4.14)

〈O〉 = 〈x|O|x〉 + 〈x|O|y〉 + 〈y|O|x〉 + 〈y|O|y〉. (4.15)

〈O〉 ∗ = 〈x|O|x〉 ∗ + 〈x|O|y〉 ∗ + 〈y|O|x〉 ∗ + 〈y|O|y〉 ∗

= 〈x|O|x〉 + 〈y|O|x〉 ∗ + 〈x|O|y〉 + 〈y|O|y〉

= 〈O〉 (4.16)

= λ (4.17)

〈ψ|O = λ〈ψ|. (4.18)

which shows that 〈ψ| is an eigenbra **of** O with real eigenvalue λ. Therefore, for an arbitrary ket,

〈ψ|O|φ〉 = λ〈ψ|φ〉 (4.19)

Now, consider eigenvectors **of** a Hermitian operator, |ψ〉 and |φ〉. Obviously we have:

Since O is Hermitian, we also have

Thus, we can write:

O|ψ〉 = λ|ψ〉 (4.20)

O|φ〉 = µ|φ〉 (4.21)

〈ψ|O = λ〈ψ| (4.22)

〈φ|O = µ〈φ| (4.23)

〈φ|O|ψ〉 = λ〈φ|ψ〉 (4.24)

〈φ|O|ψ〉 = µ〈φ|ψ〉 (4.25)

Subtracting the two: (λ − µ)〈φ|ψ〉 = 0. Thus, if λ �= µ, |ψ〉 and |φ〉 must be orthogonal.

85

4.0.3 **Quantum** Measurement:

The only possible result **of** the measurement **of** a physical quantity is one **of** the eigenvalues **of**

the corresponding observable. To any physical observable we ascribe an operator, O. The result

**of** a physical measurement must be an eigenvalue, a. With each eigenvalue, there corresponds

an eigenstate **of** O, |φa〉. This function is such that the if the state vector, |ψ(t◦)〉 = |φa〉 where

t◦ corresponds to the time at which the measurement was preformed, O|ψ〉 = a|ψ〉 and the

measurement will yield a.

Suppose the state-function **of** our system is not an eigenfunction **of** the operator we are

interested in. Using the superposition principle, we can write an arbitrary state function as a

linear combination **of** eigenstates **of** O

|ψ(t◦)〉 = �

〈φa|ψ(t◦)〉|φa〉

a

= �

ca|φa〉. (4.26)

a

where the sum is over all eigenstates **of** O. Thus, the probability **of** observing answer a is |ca| 2 .

IF the measurement DOES INDEED YIELD ANSWER a, the wavefunction **of** the system

at an infinitesmimal bit after the measurement must be in an eigenstate **of** O.

4.0.4 The Principle **of** Spectral Decomposition:

|ψ(t + ◦ )〉 = |φa〉. (4.27)

For a non-discrete spectrum: When the physical quantity, A, is measured on a system in a

normalized state |ψ〉, the probability P(an) **of** obtaining the non-degenerate eigenvalue an **of** the

corresponding observable is given by

P(an) = |〈un|ψ〉| 2

where |un〉 is a normalized eigenvector **of** A associated with the eigenvalue an. i.e.

A|un〉 = an|un〉

(4.28)

For a discrete spectrum: the sampe principle applies as in the non-discrete case, except we

sum over all possible degeneracies **of** an

gn�

P(an) = |〈un|ψ〉|

i=1

2

Finally, for the case **of** a continuous spectrum: the probability **of** obtaining a result between

α and α + dα is

dPα = |〈α|ψ〉| 2 dα

86

4.0.5 The Superposition Principle

Let’s formalize the above discussion a bit and write the electron’s state |ψ〉 = a|1〉 + b|2〉 where

|1〉 and |2〉 are “basis states” corresponding to the electron passing through slit 1 or 2. The

coefficients, a and b, are just the complex numbers ψ1 and ψ2 written above. This |ψ〉 is a vector

in a 2-dimensional complex space with unit length since ψ 2 1 + ψ 2 1 = 1. 1

Let us define a Vector Space by defining a set **of** objects {|ψ〉}, an addition rule: |φ〉 =

|ψ〉 + |ψ ′ > which allows us to construct new vectors, and a scaler multiplication rule |φ〉 = a|ψ〉

which scales the length **of** a vector. A non-trivial example **of** a vector space is the x, y plane.

Adding two vectors gives another vector also on the x, y plane and multiplying a vector by a

constant gives another vector pointed in the same direction but with a new length.

The inner product **of** two vectors is written as

〈φ|ψ〉 = (φxφy)

�

ψx

�

(4.29)

=

ψy

φ ∗ xψx + φ ∗ yψy (4.30)

= 〈ψ|φ〉 ∗ . (4.31)

The length **of** a vector is just the inner product **of** the vector with itself, i.e. ψ|ψ〉 = 1 for the

state vector we defined above.

The basis vectors for the slits can be used as a basis for an arbitrary state |ψ〉 by writing it

as a linear combination **of** the basis vectors.

|ψ〉 = ψ1|1〉 + ψ1|1〉 (4.32)

In fact, any vector in the vector space can always be written as a linear combination **of** basis

vectors. This is the superposition principle.

The different ways **of** writing the vector |ψ〉 are termed representations. Often it is easier to

work in one representation than another knowing fully that one can always switch back in forth at

will. Each different basis defines a unique representation. An example **of** a representation are the

unit vectors on the x, y plane. We can also define another orthonormal representation **of** the x, y

plane by introducing the unit vectors |r〉, |θ〉, which define a polar coordinate system. One can

write the vector v = a|x〉 + b|y > as v = √ a 2 + b 2 |r〉 + tan −1 (b/a)|θ〉 or v = r sin θ|x〉 + r cos θ|y〉

and be perfectly correct. Usually experience and insight is the only way to determine a priori

which basis (or representation) best suits the problem at hand.

Transforming between representations is accomplished by first defining an object called an

operator which has the form:

I = �

|i〉〈i|. (4.33)

The sum means “sum over all members **of** a given basis”. For the xy basis,

i

I = |x〉〈x| + |y〉〈y| (4.34)

1 The notation we are introducing here is known as “bra-ket” notation and was invented by Paul Dirac. The

vector |ψ〉 is called a “ket”. The corresponding “bra” is the vector 〈ψ| = (ψ ∗ xψ ∗ y), where the ∗ means complex

conjugation. The notation is quite powerful and we shall use is extensively throughout this course.

87

This operator is called the “idempotent” operator and is similar to multiplying by 1. For example,

We can also write the following:

I|ψ〉 = |1〉〈1|ψ〉 + |2〉〈2|ψ〉 (4.35)

= ψ1|1〉 + ψ2|2〉 (4.36)

= |ψ〉 (4.37)

|ψ〉 = |1〉〈1|ψ〉 + |2〉〈2|ψ〉 (4.38)

The state **of** a system is specified completely by the complex vector |ψ〉 which can be written

as a linear superposition **of** basis vectors spanning a complex vector space (Hilbert space). Inner

products **of** vectors in the space are as defined above and the length **of** any vector in the space

must be finite.

Note, that for state vectors in continuous representations, the inner product relation can be

written as an integral:

�

〈φ|ψ〉 = dqφ ∗ (q)φ(q) (4.39)

and normalization is given by

�

〈ψ|ψ〉 = dq|φ(q)| 2 ≤ ∞. (4.40)

The functions, ψ(q) are termed square integrable because **of** the requirement that the inner

product integral remain finite. The physical motivation for this will become apparent in a

moment when ascribe physical meaning to the mathematical objects we are defining. The class

**of** functions satisfying this requirement are also known as L 2 functions. (L is for Lebesgue,

referring to the class **of** integral.)

The action **of** the laser can also be represented mathematically as an object **of** the form

and

P1 = |1〉〈1|. (4.41)

P2 = |2〉〈2| (4.42)

and note that P1 + P2 = I.

When P1 acts on |ψ〉 it projects out only the |1〉 component **of** |ψ〉

The expectation value **of** an operator is formed by writing:

Let’s evaluate this:

P1|ψ〉 = ψ1|1〉. (4.43)

〈P1〉 = 〈ψ|P1|ψ〉 (4.44)

〈Px〉 = 〈ψ|1〉〈1|ψ〉

= ψ 2 1 (4.45)

88

Similarly for P2.

Part **of** our job is to insure that the operators which we define have physical counterparts.

We defined the projection operator, P1 = |1〉〈1|, knowing that the physical polarization filter

removed all “non” |1〉 components **of** the wave. We could have also written it in another basis,

the math would have been slightly more complex, but the result the same. |ψ1| 2 is a real number

which we presumably could set **of**f to measure in a laboratory.

4.0.6 Reduction **of** the wavepacket:

If a measurement **of** a physical quantity A on the system in the state |ψ〉 yields the result, an,

the state **of** the physical system immediately after the measurement is the normalized projection

Pn|ψ〉 onto the eigen subspace associated with an.

In more plain language, if you observe the system at x, then it is at x. This is perhaps

the most controversial posulate since it implies that the act **of** observing the system somehow

changes the state **of** the system.

Suppose the state-function **of** our system is not an eigenfunction **of** the operator we are

interested in. Using the superposition principle, we can write an arbitrary state function as a

linear combination **of** eigenstates **of** O

|ψ(t◦)〉 = �

〈φa|ψ(t◦)〉|φa〉

a

= �

ca|φa〉. (4.46)

a

where the sum is over all eigenstates **of** O. Thus, the probability **of** observing answer a is |ca| 2 .

IF the measurement DOES INDEED YIELD ANSWER a, the wavefunction **of** the system

at an infinitesmimal bit after the measurement must be in an eigenstate **of** O.

|ψ(t + ◦ )〉 = |φa〉. (4.47)

This is the only postulate which is a bit touchy deals with the reduction **of** the wavepacket

as the result **of** a measurement. On one hand, you could simply accept this as the way one

goes about business and simply state that quantum mechanics is an algorithm for predicting

the outcome **of** experiments and that’s that. It says nothing about the inner workings **of** the

universe. This is what is known as the “Reductionist” view point. In essence, the Reductionist

view point simply wants to know the answer: “How many?”, “How wide?”, “How long?”.

On the other hand, in the Holistic view, quantum mechanics is the underlying physical theory

**of** the universe and say that the process **of** measurement does play an important role in how the

universe works. In otherwords, in the Holist wants the (w)hole picture.

The Reductionist vs. Holist argument has been the subject **of** numerous articles and books

in both the popular and scholarly arenas. We may return to the philosophical discussion, but

for now we will simply take a reductionist view point and first learn to use quantum mechanics

as a way to make physical predictions.

89

4.0.7 The temporal evolution **of** the system:

The time evolution **of** the state vector is given by the Schrödinger equation

i¯h ∂

|ψ(t)〉 = H(t)|ψ(t)〉

∂t

where H(t) is an operator/observable associated withthe total energy **of** the system.

As we shall see, H is the Hamiltonian operator and can be obtained from the classical Hamiltonian

**of** the system.

4.0.8 Dirac **Quantum** Condition

One **of** the crucial aspects **of** any theory is that we need to be able to construct physical observables.

Moreover, we would like to be able to connect the operators and observables in quantum

mechanics to the observables in classical mechanics. At some point there must be a correspondence.

This connection can be made formally by relating what is known as the Poisson bracket

in classical mechanics:

{f(p, q), g(p, q)} = ∂f ∂g ∂g ∂f

−

∂q ∂p ∂q ∂p

which looks a lot like the commutation relation between two linear operators:

(4.48)

[ Â, ˆ B] = Â ˆ B − ˆ BÂ (4.49)

Of course, f(p, q) and g(p, q) are functions over the classical position and momentum **of** the

physical system. For position and momentum, it is easy to show that the classical Poisson

bracket is

{q, p} = 1.

Moreover, the quantum commutation relation between the observable x and p is

[ˆx, ˆp] = i¯h.

Dirac proposed that the two are related and that this relation defines an acceptible set **of**

quantum operations.

The quantum mechanical operators ˆ f and ˆg, which in classical theory replace the

classically defined functions f and g, must always be such that the commutator **of** ˆ f

and ˆg corresponds to the Poisson bracket **of** f and g according to

To see how this works, we write the momentum operator as

i¯h{f, g} = [ ˆ f, ˆg] (4.50)

ˆp = ¯h

i

∂

∂x

90

(4.51)

Thus,

Thus,

Let’s see if ˆx and ˆp commute. First **of** all

ˆpψ(x) = ¯h ∂ψ(x)

i ∂x

(4.52)

∂

∂x xf(x) = f(x) + xf ′ (x) (4.53)

[ˆx, ˆp]f(x) = ¯h ∂ ∂

(x f(x) −

i ∂x ∂x xf(x)

= ¯h

i (xf ′ (x) − f(x) − xf ′ (x))

= i¯hf(x) (4.54)

The fact that ˆx and ˆp do not commute has a rather significant consequence:

In other words, if two operators do not commute, one cannot devise and experiment to

simultaneously measure the physical quantities associated with each operator. This in fact limits

the precision in which we can preform any physical measurement.

The principle result **of** the postulates is that the wavefunction or state vector **of** the system

carries all the physical information we can obtain regarding the system and allows us to make

predictions regarding the probable outcomes **of** any experiment. As you may well know, if one

make a series **of** experimental measurements on identically prepared systems, one obtains a

distribution **of** results–usually centered about some peak in the distribution.

When we report data, we usually don’t report the result **of** every single experiment. For

a spectrscopy experiment, we may have made upwards **of** a million individual measurement,

all distributed about some average value. From statistics, we know that the average **of** any

distribution is the expectation value **of** some quantity, in this case x:

for the case **of** a discrete spectra, we would write

�

E(x) = P(x)xdx (4.55)

E[h] = �

n

hnPn

where hn is some value and Pn the number **of** times you got that value normalized so that

�

Pn = 1

n

. In the language above, the hn’s are the possible eigenvalues **of** the h operator.

A similar relation holds in quantum mechanics:

(4.56)

Postulate 4.1 Observable quantities are computed as the expectation value **of** an operator 〈O〉 =

〈ψ|O|ψ〉. The expectation value **of** an operator related to a physical observable must be real.

91

For example, the expectation value **of** ˆx the position operator is computed by the integral

or for the discrete case:

〈x〉 =

� +∞

−∞

ψ ∗ (x)xψ(x)dx.

〈O〉 = �

on|〈n|ψ〉| 2 .

n

Of course, simply reporting the average or expectation values **of** an experiment is not enough,

the data is usually distributed about either side **of** this value. If we assume the distribution is

Gaussian, then we have the position **of** the peak center xo = 〈x〉 as well as the width **of** the

Gaussian σ 2 .

The mean-squared width or uncertainty **of** any measurement can be computed by taking

σ 2 A = 〈(A − 〈A〉)〉.

In statistical mechanics, this the fluctuation about the average **of** some physical quantity, A. In

quantum mechanics, we can push this definition a bit further.

Writing the uncertainty relation as

σ 2 A = 〈(A − 〈A〉)(A − 〈A〉)〉 (4.57)

= 〈ψ|(A − 〈A〉)(A − 〈A〉)|ψ〉 (4.58)

= 〈f|f〉 (4.59)

where the new vector |f〉 is simply short hand for |f〉 = (A − 〈A〉)|ψ〉. Likewise for a different

operator B

σ 2 B = = 〈ψ|(B − 〈B〉)(B − 〈B〉)|ψ〉 (4.60)

We now invoke what is called the Schwartz inequality

= 〈g|g〉. (4.61)

σ 2 Aσ 2 B = 〈f|f〉〈g|g〉 ≥ |〈f|g〉| 2

So if we write 〈f|g〉 as a complex number, then

So we conclude

|〈f|g〉| 2 = |z| 2

σ 2 Aσ 2 B ≥

= ℜ(z) 2 + ℑ(z) 2

≥ ℑ(z) 2 �

1

=

2i (z − z∗ )

� �2

1

≥ (〈f|g〉 − 〈g|f〉)

2i

� �2

1

(〈f|g〉 − 〈g|f〉)

2i

92

�2

(4.62)

(4.63)

(4.64)

Now, we reinsert the definitions **of** |f〉 and |g〉.

Likewise

Combining these results, we obtain

〈f|g〉 = 〈ψ|(A − 〈A〉)(B − 〈B〉)|ψ〉

= 〈ψ|(AB − 〈A〉B − A〈B〉 + 〈A〉〈B〉)|ψ〉

= 〈ψ|AB|ψ〉 − 〈A〉〈ψ|B|ψ〉 − 〈B〉〈ψ|A|ψ〉 + 〈A〉〈B〉

= 〈AB〉 − 〈A〉〈B〉 (4.65)

〈g|f〉 = 〈BA〉 − 〈A〉〈B〉. (4.66)

〈f|g〉 − 〈g|f〉 = 〈AB〉 − 〈BA〉 = 〈AB − BA〉 = 〈[A, B]〉. (4.67)

So we finally can conclude that the general uncertainty product between any two operators is

given by

σ 2 Aσ 2 � �2

1

B = 〈[A, B]〉

(4.68)

2i

This is commonly referred to as the Generalized Uncertainty Principle. What is means is that

for any pair **of** observables whose corresponding operators do not commute there will always be

some uncertainty in making simultaneous measurements. In essence, if you try to measure two

non-commuting properties simultaneously, you cannot have an infinitely precise determination **of**

both. A precise determination **of** one implies that you must give up some certainty in the other.

In the language **of** matrices and linear algebra this implies that if two matrices do not commute,

then one can not bring both matrices into diagonal form using the same transformation

matrix. in other words, they do not share a common set **of** eigenvectors. Matrices which do

commute share a common set **of** eigenvectors and the transformation which diagonalizes one will

also diagonalize the other.

Theorem 4.1 If two operators A and B commute and if |ψ〉 is an eigenvector **of** A, then B|ψ〉

is also an eigenvector **of** A with the same eigenvalue.

Pro**of**: If |ψ〉 is an eigenvector **of** A, then A|ψ〉 = a|ψ〉. Thus,

Assuming A and B commute, i.e. [A, B] = AB − BA = 0,

Thus, (B|ψ〉) is an eigenvector **of** A with eigenvalue, a.

BA|ψ〉 = aB|ψ〉 (4.69)

AB|ψ〉 = a(B|ψ〉) (4.70)

Exercise 4.1 1. Show that matrix multiplication is associative, i.e. A(BC) = (AB)C, but

not commutative (in general), i.e. BC �= CB

2. Show that (A + B)(A − B) = A 2 + B 2 only **of** A and B commute.

3. Show that if A and B are both Hermitian matrices, AB + BA and i(AB − BA) are also

Hermitian. Note that Hermitian matrices are defined such that Aij = A ∗ ji where ∗ denotes

complex conjugation.

93

4.1 Dirac Notation and Linear Algebra

Part **of** the difficulty in learning quantum mechanics comes from fact that one must also learn a

new mathematical language. It seems very complex from the start. However, the mathematical

objects which we manipulate actually make life easier. Let’s explore the Dirac notation and the

related mathematics.

We have stated all along that the physical state **of** the system is wholly specified by the

state-vector |ψ〉 and that the probability **of** finding a particle at a given point x is obtained via

|ψ(x)| 2 . Say at some initial time |ψ〉 = |s〉 where s is some point along the x axis. Now, the

amplitude to find the particle at some other point is 〈x|s〉. If something happens between the

two points we write

〈x|operator describing process|s〉 (4.71)

The braket is always read from right to left and we interpret this as the amplitude for“starting

**of**f at s, something happens, and winding up at i”. An example **of** this is the Go function in the

homework. Here, I ask “what is the amplitude for a particle to start **of**f at x and to wind up at

x ′ after some time interval t?”

Another Example: Electrons have an intrinsic angular momentum called “spin” . Accordingly,

they have an associated magnetic moment which causes electrons to align with or against an

imposed magnetic field (eg.this gives rise to ESR). Lets say we have an electron source which

produces spin up and spin down electrons with equal probability. Thus, my initial state is:

|i〉 = a|+〉 + b|−〉 (4.72)

Since I’ve stated that P (a) = P (b), |a| 2 = |b| 2 . Also, since P (a) + P (b) = 1, a = b = 1/ √ 2.

Thus,

|i〉 = 1

√ 2 (|+〉 + |−〉) (4.73)

Let’s say that the spin ups can be separated from the spin down via a magnetic field, B and we

filter **of**f the spin down states. Our new state is |i ′ 〉 and is related to the original state by

4.1.1 Transformations and Representations

〈i ′ |i〉 = a〈+|+〉 + b〈+|−〉 = a. (4.74)

If I know the amplitudes for |ψ〉 in a representation with a basis |i〉 , it is always possible to

find the amplitudes describing the same state in a different basis |µ〉. Note, that the amplitude

between two states will not change. For example:

also

|a〉 = �

|i〉〈i|a〉 (4.75)

i

|a〉 = �

|µ〉〈µ|a〉 (4.76)

µ

94

Therefore,

and

〈µ|a〉 = �

〈µ|i〉〈i|a〉 (4.77)

i

〈i|a〉 = �

〈i|µ〉〈µ|a〉. (4.78)

µ

Thus, the coefficients in |µ〉 are related to the coefficients in |i〉 by 〈µ|i〉 = 〈i|µ〉 ∗ . Thus, we can

define a transformation matrix Sµi as

and a set **of** vectors

Thus, we can see that

Thus,

Now, we can also write

Siµ =

Sµi =

⎡

⎢

⎣

⎡

⎢

⎣

〈µ|i〉 〈µ|j〉 〈µ|k〉

〈ν|i〉 〈ν|j〉 〈ν|k〉

〈λ|i〉 〈λ|j〉 〈λ|k〉

ai =

aµ =

⎡

⎢

⎣

⎡

⎢

⎣

aµ = �

〈i|a〉

〈j|a〉

〈k|a〉

〈µ|a〉

〈ν|a〉

〈λ|a〉

i

⎤

Sµiai

⎤

⎥

⎦ (4.79)

⎥

⎦ (4.80)

⎤

〈µ|i〉 ∗ 〈µ|j〉 ∗ 〈µ|k〉 ∗

〈ν|i〉 〈ν|j〉 〈ν|k〉 ∗

〈λ|i〉 ∗ 〈λ|j〉 ∗ 〈λ|k〉 ∗

ai = �

µ

Siµaµ

⎥

⎦ (4.81)

⎤

⎥

⎦ = S ∗ µi

Since 〈i|µ〉 = 〈µ|i〉 ∗ , S is the Hermitian conjugate **of** S. So we write

S = S †

95

(4.82)

(4.83)

(4.84)

(4.85)

(4.86)

So in short;

thus

S † = (S T ) ∗

(S † )ij = S ∗ ji

and S is called a unitary transformation matrix.

4.1.2 Operators

(4.87)

(4.88)

(4.89)

a = Sa = SSa = S † Sa (4.90)

S † S = 1 (4.91)

A linear operator Â maps a vector in space H on to another vector in the same space. We can

write this in a number **of** ways:

or

Linear operators have the property that

|φ〉 Â

↦→ |χ〉 (4.92)

|χ〉 = Â|φ〉 (4.93)

Â(a|φ〉 + b|χ〉) = aÂ|φ〉 + bÂ|χ〉 (4.94)

Since superposition is rigidly enforced in quantum mechanics, all QM operators are linear operators.

The Matrix Representation **of** an operator is obtained by writing

Aij = 〈i| Â|j〉 (4.95)

For example: Say we know the representation **of** A in the |i〉 basis, we can then write

Thus,

�

|χ〉 = Â|φ〉 = Â|i〉〈i|φ〉 = �

|j〉〈j|χ〉 (4.96)

i

j

〈j|χ〉 = �

〈j|A|i〉〈i|φ〉 (4.97)

We can keep going if we want by continuing to insert 1’s where ever we need.

i

96

The matrix A is Hermitian if A = A † . If it is Hermitian, then I can always find a basis |µ〉 in

which it is diagonal, i.e.

So, what is Â|µ〉 ?

Aµν = aµδµν

(4.98)

Â|µ〉 = �

|i〉〈i|A|j〉〈j|µ〉 (4.99)

ij

= �

|i〉Aijδjµ

ij

= �

|i〉Aiµ

i

= �

|i〉aµδiµ

i

(4.100)

(4.101)

(4.102)

(4.103)

(4.104)

(4.105)

(4.106)

= aµ|µ〉 (4.107)

An important example **of** this is the “time-independent” Schroedinger Equation:

which we spend some time in solving above.

Finally, if Â|φ〉 = |χ〉 then 〈φ|A† = 〈χ|.

4.1.3 Products **of** Operators

An operator product is defined as

ˆH|ψ〉 = E|ψ〉 (4.108)

( Â ˆ B)|ψ〉 = Â[ ˆ B|ψ〉] (4.109)

where we operate in order from right to left. We proved that in general the ordering **of** the

operations is important. In other words, we cannot in general write Â ˆ B = ˆ BÂ. An example **of**

this is the position and momentum operators. We have also defined the “commutator”

[ Â, ˆ B] = Â ˆ B − ˆ BÂ. (4.110)

Let’s now briefly go over how to perform algebraic manipulations using operators and commutators.

These are straightforward to prove

97

1. [ Â, ˆ B] = −[ ˆ B, Â]

2. [ Â, Â] = −[Â, Â] = 0

3. [ Â, ˆ BĈ] = [Â, ˆ B] Ĉ + ˆ B[ Â, Ĉ]

4. [ Â, ˆ B + Ĉ] = [Â, ˆ B] + [ Â, Ĉ]

5. [ Â, [ ˆ B, Ĉ]] + [ ˆ B, [ Ĉ, Â]] + [Ĉ, [Â, ˆ B]] = 0 (Jacobi Identity)

6. [ Â, ˆ B] † = [ Â† , ˆ B † ]

4.1.4 Functions Involving Operators

Another property **of** linear operators is that the inverse operator always can be found. I.e. if

|χ〉 = Â|φ〉 then there exists another operator ˆ B such that |φ〉 = ˆ B|χ〉. In other words ˆ B = Â−1 .

We also need to know how to evaluate functions **of** operators. Say we have a function, F (z)

which can be expanded as a series

Thus, by analogy

For example, take exp( Â)

thus

exp(x) =

∞�

F (z) = fnz

n=0

n

F ( Â) =

∞�

n=0

(4.111)

∞�

fnÂn . (4.112)

n=0

x n

n! = 1 + x + x2 /2 + · · · (4.113)

exp( Â) =

If Â is Hermitian, then F (Â) is also Hermitian. Also, note that

Likewise, if

then

∞�

n=0

Â n

n!

[ Â, F (Â)] = 0.

(4.114)

Â|φa〉 = a|φa〉 (4.115)

Â n |φa〉 = a n |φa〉. (4.116)

98

Thus, we can show that

F ( Â)|φa〉 = �

n

fn Ân |φa〉 (4.117)

(4.118)

= �

fna n |φa〉 (4.119)

n

(4.120)

= F (a) (4.121)

Note, however, that care must be taken when we evaluate F ( Â + ˆ B) if the two operators

do not commute. We ran into this briefly in breaking up the propagator for the Schroedinger

Equation in the last lecture (Trotter Product). For example,

exp( Â + ˆ B) �= exp( Â) exp( ˆ B) (4.122)

unless [ Â, ˆ B] = 0. One can derive, however, a useful formula (Glauber)

exp( Â + ˆ B) = exp( Â) exp( ˆ B) exp(−[ Â, ˆ B]/2) (4.123)

Exercise 4.2 Let H be the Hamiltonian **of** a physical system and |φn〉 the solution **of**

1. For an arbitrary operator, Â, show that

2. Let

(a) Compute [H, ˆp], [H, ˆx], and [H, ˆxˆp].

(b) Show 〈φn|ˆp|φn〉 = 0.

H|φn〉 = En|φn〉 (4.124)

〈φn|[ Â, H]|φn〉 = 0 (4.125)

H = 1

2m ˆp2 + V (ˆx) (4.126)

(c) Establish a relationship between the average **of** the kinetic energy given by

and the average force on a particle given by

Ekin = 〈φn| ˆp2

2m |φn〉 (4.127)

∂V (x)

F = 〈φn|ˆx

∂x |φn〉. (4.128)

Finally, relate the average **of** the potential for a particle in state |φn〉 to the average

kinetic energy.

99

Exercise 4.3 Consider the following Hamiltonian for 1d motion with a potential obeying a simple

power-law

H = p2

+ αxn

2m

where α is a constant and n is an integer. Calculate

(4.129)

〈A〉 = 〈ψ|[xp, H]||ψ〉 (4.130)

and use the result to relate the average potential energy to the average kinetic energy **of** the

system.

4.2 Constants **of** the Motion

In a dynamical system (quantum, classical, or otherwise) a constant **of** the motion is any quantity

such that

For quantum systems, this means that

∂tA = 0. (4.131)

[A, H] = 0 (4.132)

(What’s the equivalent relation for classical systems?) In other words, any quantity which

commutes with H is a constant **of** the motion. Furthermore, for any conservative system (in

which there is no net flow **of** energy to or from the system),

From Eq.4.131, we can write that

[H, H] = 0. (4.133)

∂t〈A〉 = ∂t〈ψ(t)|A|ψ(t)〉 (4.134)

Since [A, H] = 0, we know that if the state |φn〉 is an eigenstate **of** H,

then

H|φn〉 = En|φn〉 (4.135)

A|φn〉 = an|φn〉 (4.136)

The an are **of**ten referred to as “good quantum numbers”. What are some constants **of** the

motion for systems that we have studied thus far? (Bonus: how are constants **of** motion related

to particular symmetries **of** the system?)

A state which is in an eigenstate **of** H it’s also in an eigenstate **of** A. Thus, I can simultaneously

measure quantities associated with H and A. Also, after I measure with A, the system remains

in a the original state.

100

4.3 Bohr Frequency and Selection Rules

What if I have another operator, B, which does not commute with H? What is 〈B(t)〉? This we

can compute by first writing

|ψ(t)〉 = �

cne −iEnt/¯h |φn〉. (4.137)

Then

= �

Let’s define the “Bohr Frequency” as ωnm = (En − Em)/¯h.

〈B(t)〉 = �

n

n

〈B(t)〉 = 〈ψ|B|ψ(t)〉 (4.138)

cnc ∗ me −i(En−Em)t/¯h 〈φm|B|φn〉. (4.139)

n

cnc ∗ me −iωnmt 〈φm|B|φn〉. (4.140)

Now, the observed expectation value **of** B oscillates in time at number **of** frequencies corresponding

to the energy differences between the stationary states. The matrix elements Bnm =

〈φm|B|φn〉 do not change with time. Neither do the coefficients, {cn}. Thus, let’s write

B(ω) = cnc ∗ m〈φm|B|φn〉δ(ω − ωnm) (4.141)

and transform the discrete sum into an continuous integral

〈B(t)〉 = 1

� ∞

e

2π 0

−iωt B(ω) (4.142)

where B(ω) is the power spectral **of** B. In other words, say I monitor < B(t) > with my

instrument for a long period **of** time, then take the Fourier Transform **of** the time-series. I get

the power-spectrum. What is the power spectrum for a set **of** discrete frequencies: If I observe

the time-sequence for an infinite amount **of** time, I will get a series **of** discretely spaced sticks

along the frequency axis at precisely the energy difference between the n and m states. The

intensity is related to the probability **of** making a transition from n to m under the influence

**of** B. Certainly, some transitions will not be allowed because 〈φn|B|φm〉 = 0. These are the

“selection rules”.

We now prove an important result regarding the integrated intensity **of** a series **of** transitions:

Exercise 4.4 Prove the Thomas-Reiche-Kuhn sum rule: 2

� 2m|xn0| 2

n

¯h 2 (En − Eo) = 1 (4.143)

where the sum is over a compete set **of** states, |ψn〉 **of** energy En **of** a particle **of** mass m which

moves in a potential; |ψo〉 represents a bound state, and xn0 = 〈ψn|x|ψo〉. (Hint: use the commutator

identity: [x, [x, H]] = ¯h 2 /m)

2 This is perhaps one **of** the most important results **of** quantum mechanics since it is gives the total spectral

intensity for a series **of** transitions. c.f Bethe and Jackiw for a great description **of** sum-rules.

101

Figure 4.4: The diffraction function sin(x)/x

1

0.8

0.6

0.4

0.2

-20 -10 10 20

-0.2

4.4 Example using the particle in a box states

What are the constants **of** motion for a particle in a box?

Recall that the energy levels and wavefunctions for this system are

φn(x) =

2ma2 (4.144)

�

2

a sin(nπ x) (4.145)

a

En = n2 π 2 ¯h 2

Say our system in in the nth state. What’s the probability **of** measuring the momentum and

obtaining a result between p and p + dp?

where

Pn(p)dp = |φn(p)| 2 dp (4.146)

�

�

1 a 2

φn(p) = √ dx

2π¯h 0 a sin(nπx/a)e−ipx/¯h

(4.147)

= 1

�

i(nπ/a−p/¯h)a 1 e − 1

√

2i 2π¯h i(nπ/a − p/¯h) − e−i(nπ/a−p/¯h)a �

− 1

(4.148)

−i(nπ/a − p/¯h)

= 1

�

a

2i π¯h exp i(nπ/2 − pa/(2¯h))[F (p − nπ¯h/2) + (−1)n+1F (p + nπ¯h/a)] (4.149)

Where the F (p) are “diffraction functions”

F (p) = sin(pa/(2¯h))

pa/(2¯h)

(4.150)

Note that the width 4π¯h/a does not change as I change n. Nor does the amplitude. However,

note that (F (x + n) ± FF (x − n)) 2 is always an even function **of** x. Thus, we can say

〈p〉n =

� +∞

−∞

Pn(p)pdp = 0 (4.151)

102

We can also compute:

〈p 2 〉 = ¯h 2

� � �

∞ �

�∂φn(x)

�2

�

dx � �

0 � ∂x �

= ¯h 2

� a

0

=

2

a

(4.152)

� �

nπ 2

cos(nπx/a)dx (4.153)

a

� nπ¯h

a

� 2

Thus, the RMS deviation **of** the momentum:

∆pn =

= 2mEn

�

〈p 2 〉n − 〈p〉 2 n = nπ¯h

a

(4.154)

(4.155)

Thus, as n increases, the relative accuracy at which we can measure p increases due the fact

that we can resolve the wavefunction into two distinct peaks corresponding to the particle either

going to the left or to the right. ∆p increases due to the fact that the two possible choices for

the measurement are becoming farther and farther apart and hence reflects the distance between

the two most likely values.

4.5 Time Evolution **of** Wave and Observable

Now, suppose we put our system into a superposition **of** box-states:

|ψ(0)〉 = 1

√ 2 (|φ1〉 + |φ2〉) (4.156)

What is the time evolution **of** this state? We know the eigen-energies, so we can immediately

write:

|ψ(t)〉 = 1

√ 2 (exp(−iE1t/¯h)|φ1〉 + exp(−iE2t/¯h)|φ2〉) (4.157)

Let’s factor out a common phase factor **of** e −iE1t/¯h and write this as

|ψ(t)〉 ∝ 1

√ 2 (|φ1〉 + exp(−i(E2 − E1)t/¯h)|φ2〉) (4.158)

and call (E2 − E1)/¯h = ω21 the Bohr frequency.

where

|ψ(t)〉 ∝ 1

√ 2 (|φ1〉 + exp(−iω21t)|φ2〉) (4.159)

ω21 = 3π2¯h . (4.160)

2ma2 103

The phase factor is relatively unimportant and cancels out when I make a measurement. Eg.

the prob. density:

|ψ(x, t)| 2 = |〈x|ψ(t)〉| 2

(4.161)

(4.162)

= 1

2 φ2 1(x) + 1

2 φ2 2(x) + φ1(x)φ2(x) cos(ω21t) (4.163)

Now, let’s compute 〈x(t)〉 for the two state system. To do so, let’s first define x ′ = x − a/2

as the center **of** the well to make the integrals easier. The first two are easy:

〈φ1|x ′ |φ1〉 ∝

〈φ2|x ′ |φ2〉 ∝

which we can do by symmetry. Thus,

Thus,

� a

0

� a

0

dx(x − a/2) sin 2 (πx/a) = 0 (4.164)

dx(x − a/2) sin 2 (2πx/a) = 0 (4.165)

〈x ′ (t)〉 = Re{e −iω21t 〈φ1|x ′ |φ2〉} (4.166)

〈φ1|x ′ |φ2〉 = 〈φ1|x|φ2〉 − (a/2)〈φ1|φ2〉 (4.167)

= 2

� a

dxx sin(πx/a) sin(2πx/a) (4.168)

a 0

= − 16a

9π2 (4.169)

〈x(t)〉 = a 16a

−

2 9π2 cos(ω21t) (4.170)

Compare this to the classical trajectory. Also, what about 〈E(t)〉?

4.6 “Unstable States”

So far in this course, we have been talking about systems which are totally isolated from the

rest **of** the universe. In these systems, there is no influx or efflux **of** energy and all our dynamics

are governed by the three principle postulates I mentioned a the start **of** the lecture. IN essence,

if at t = 0 I prepare the system in an eigenstate **of** H, then for all times later, it’s still in that

state (to within a phase factor). Thus, in a strictly conservative system, a system prepared in

an eigenstate **of** H will remain in an eigenstate forever.

However, this is not exactly what is observed in nature. We know from experience that atoms

and molecules, if prepared in an excited state (say via the absorption **of** a photon) can relax

104

ack to the ground state or some lower state via the emission **of** a photon or a series **of** photons.

Thus, these eigenstates are “unstable”.

What’s wrong here? The problem is not so much to do with what is wrong with our description

**of** the isolated system, it has to do with full description is not included. A isolated atom or

molecule can still interact with the electro-magnetic field (unless we do some tricky confinement

experiments). Thus, there is always some interaction with an outside “environment”. Thus,

while it is totally correct to describe the evolution **of** the global system in terms **of** some global

“atom” + “environment” Hamiltonian, it it NOT totally rigorous to construct a Hamiltonian

which describes only part **of** the story. But, as the great **Pr of**. Karl Freed (at U. Chicago) once

told me “Too much rigor makes rigor mortis”.

Thankfully, the coupling between an atom and the electromagnetic field is pretty weak. Each

photon emission probability is weighted by the fine-structure constant, α ≈ 1/137. Thus a 2

photon process is weighted by α 2 . Thus, the isolated system approximation is pretty good. Also,

we can pretty much say that most photon emission processes occur as single photon events.

Let’s play a bit “fast and loose” with this idea. We know from experience that if we prepare

the system in an excited state at t = 0, the probability **of** finding it still in the excited state at

some time t later, is

P (t) = e −t/τ

(4.171)

where τ is some time constant which we’ll take as the lifetime **of** the state. One way to “prove”

this relation is to go back to Problem Set 0. Let’s say we have a large number **of** identical systems

N , each prepared in the excited state at t = 0. At time t, there are

N(t) = N e −t/τ

(4.172)

systems in the excited state. Between time t and t + dt a certain number, dn(t) will leave the

excited state via photon emission.

Thus,

dn(t) = N(t) − N(t + dt) = − dN(t)

dt = N(t)dt

dt τ

dn(t)

N(t)

= dt

τ

Thus, 1/τ is the probability per unit time for leaving the unstable state.

The Avg. time a system spends in the unstable state is given by:

1

τ

(4.173)

(4.174)

� ∞

dtte

0

−t/τ = τ (4.175)

For a stable state P (t) = 1 thus, τ → ∞.

The time a system spends in the state is independent **of** its history. This is a characteristic **of**

an unstable state. (Also has to do with the fact that the various systems involved to not interact

with each other. )

105

Finally, according to the time-energy uncertainty relation:

∆Eτ ≈ ¯h. (4.176)

Thus, an unstable system has an intrinsic “energy width” associated with the finite time the

systems spends in the state.

For a stable state:

and

for real energies.

What if I instead write: E ′ n = En − i¯hγn/¯h? Then

Thus,

|ψ(t)〉 = e −iEnt/¯h |φn〉 (4.177)

Pn(t) = |e −iEnt/¯h | 2 = 1 (4.178)

Pn(t) = |e −iEnt/¯h e −γn/2t | 2 = e −γnt

γn = 1/τn

(4.179)

(4.180)

is the “Energy Width” **of** the unstable state.

The surprising part **of** all this is that in order to include dissipative effects (photon emission,

etc..) the Eigenvalues **of** H become complex. In other words, the system now evolves under a

non-hermitian Hamiltonian! Recall the evolution operator for an isolated system:

U(t) = e −iHt/¯h (4.181)

(4.182)

U † (t) = e iHt/¯h (4.183)

where the first is the forward evolution **of** the system and the second corresponds to the backwards

evolution **of** the system. Thus, Unitarity is thus related to the time-reversal symmetry

**of** conservative systems. The inclusion **of** an “environment” breaks the intrinsic time-reversal

symmetry **of** an isolated system.

4.7 Problems and Exercises

Exercise 4.5 Find the eigenvalues and eigenvectors **of** the matrix:

⎛

⎜

M = ⎜

⎝

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

106

⎞

⎟

⎠

(4.184)

Solution: You can either do this the hard way by solving the secular determinant and then

finding the eigenvectors by Gramm-Schmidt orthogonalization, or realize that since M = M −1

and M = M † , M is a unitary matrix, this, its eigenvalues can only be ± 1. Furthermore, since

the trace **of** M is 0, the sum **of** the eigenvalues must be 0 as well. Thus, λ = (1, 1, −1, −1) are

the eigenvalues. To get the eigenvectors, consider the following. Let φmu be an eigenvector **of**

M, thus,

⎛

⎜

φµ = ⎜

⎝

Since Mφµ = λµφmu, x1 = λµx4 and x2 = λµx3 Thus, 4 eigenvectors are

for λ = (−1, −1, 1, 1).

⎛

⎜

⎝

−1

0

0

1

⎞

⎛

⎟

⎠ ,

⎜

⎝

0

1

−1

0

⎞

x1

x2

x3

x4

⎛

⎟

⎠ ,

⎜

⎝

Exercise 4.6 Let λi be the eigenvalues **of** the matrix:

calculate the sums:

and

H =

⎛

⎜

⎝

⎞

⎟ . (4.185)

⎠

1

0

0

1

⎞

2 −1 −3

−1 1 2

−3 2 3

3�

λi

i

3�

λ

i

2 i

⎛

⎟

⎠ ,

⎜

⎝

⎞

0

1

1

0

⎞

⎟

⎠

(4.186)

⎟

⎠ (4.187)

Hint: use the fact that the trace **of** a matrix is invariant to choice representation.

Solution: Using the hint,

and

(4.188)

(4.189)

trH = �

λi = �

Hii = 2 + 3 + 1 = 6 (4.190)

i

i

�

λ

i

2 i = �

HijHji =

ij

�

H

ij

2 ij = 42 (4.191)

107

Exercise 4.7 1. Let |φn〉 be the eigenstates **of** the Hamiltonian, H **of** some arbitrary system

which form a discrete, orthonormal basis.

Define the operator, Unm as

(a) Calculate the adjoint U † nm **of** Unm.

(b) Calculate the commutator, [H, Unm].

(c) Prove: UmnU † pq = δnqUmp

H|φn〉 = En|φn〉.

Unm = |φn〉〈φm|.

(d) For an arbitrary operator, A, prove that

〈φn|[A, H]|φn〉 = 0.

(e) Now consider some arbitrary one dimensional problem for a particle **of** mass m and

potential V (x). For here on, let

H = p2

+ V (x).

2m

i. In terms **of** p, x, and V (x), compute: [H, p], [H, x], and [H, xp].

ii. Show that 〈φn|p|φn〉 = 0.

iii. Establish a relation between the average value **of** the kinetic energy **of** a state

(f) Show that

〈T 〉 = 〈φn| p2

2m |φn〉

and

〈φn|x dV

dx |φn〉.

The average potential energy in the state φn is

〈V 〉 = 〈φn|V |φn〉,

find a relation between 〈V 〉 and 〈T 〉 when V (x) = Vox λ for λ = 2, 4, 6, . . ..

〈φn|p|φm〉 = α〈φn|x|φm〉

where α is some constant which depends upon En − Em. Calculate α, (hint, consider

the commutator [x, H] which you computed above.

(g) Deduce the following sum-rule for the linear -response function.

〈φ0|[x, [H, x]]|φ0〉 = 2 �

(En − E0)|〈φ0|x|φn〉| 2

n>0

Here |φ0〉 is the ground state **of** the system. Give a physical interpretation **of** this last

result.

108

Exercise 4.8 For this section, consider the following 5 × 5 matrix:

⎛

⎜

H = ⎜

⎝

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 1 0 0 0

1 0 0 0 0

⎞

⎟

⎠

(4.192)

1. Using Mathematica determine the eigenvalues, λj, and eigenvectors, φn, **of** H using the

Eigensystem[] command. Determine the eigenvalues only by solving the secular determinant.

|H − Iλ| = 0

Compare the computational effort required to perform both calculations. Note: in entering

H into Mathematica, enter the numbers as real numbers rather than as integers (i.e. 1.0

vs 1 ).

2. Show that the column matrix **of** the eigenvectors **of** H,

T = {φ1, . . . , φ5},

provides a unitary transformation **of** H between the original basis and the eigenvector basis.

T † HT = Λ

where Λ is the diagonal matrix **of** the eigenvalues λj. i.e. Λij = λiδij.

3. Show that the trace **of** a matrix is invarient to representation.

4. First, without using Mathematica, compute: T r(H 2 ). Now check your result with Mathematica.

109

Chapter 5

Bound States **of** The Schrödinger

Equation

A #2 pencil and a dream can take you anywhere.

– Joyce A. Myers

Thus far we have introduced a series **of** postulates and discussed some **of** their physical implications.

We have introduced a powerful notation (Dirac Notation) and have been studying how

we describe dynamics at the atomic and molecular level where ¯h is not a small number, but is

**of** order unity. We now move to a topic which will serve as the bulk **of** our course, the study

**of** stationary systems for various physical systems. We shall start with some general principles,

(most **of** which we have seen already), and then tackle the following systems in roughly this

order:

1. Harmonic Oscillators: Molecular vibrational spectroscopy, phonons, photons, equilibrium

quantum dynamics.

2. Angular Momentum: Spin systems, molecular rotations.

3. Hydrogen Atom: Hydrogenic Systems, basis for atomic theory

5.1 Introduction to Bound States

Before moving on to these systems, let’s first consider what is meant by a “bound state”. Say

we have a potential well which has an arbitrary shape except that at x = ±a, V (x) = 0 and

remains so in either direction. Also, in the range **of** −a ≤ x ≤ a, V (x) < 0. The Schrodinger

Equation for the stationary states is:

� −¯h 2

2m

∂2 �

+ V (x)

∂x2 φn(x) = Enφn(x) (5.1)

Rather than solve this exactly (which we can not do since we haven’t specified more about

V ) let’s examine the topology **of** the allowed bound state solutions. As we have done with the

square well cases, let’s cut the x axis into three domains: Domain 1 for x < −a, Domain 2 for

−a ≤ x ≤ a, Domain 3 for x > a. What are the matching conditions that must be met?

110

For Domain 1 we have:

For Domain 2 we have:

For Domain 3 we have:

∂2 ∂x2 φI(x) = − 2mE

¯h 2 φI(x) ⇒ (+)φI(x) (5.2)

∂ 2

∂x 2 φII(x) =

2m(V (x) − E)

¯h 2 φII(x) ⇒ (−)φII(x) (5.3)

∂2 ∂x2 φIII(x) = − 2m(E)

¯h 2 φIII(x) ⇒ (+)φIII(x) (5.4)

At the rightmost end **of** each equation, the (±) indicates the sign **of** the second derivative **of**

the wavefunction. (i.e. the curvature must have the same or opposite sign as the function itself.)

For the + curvature functions, the wavefunctions curve away from the x-axis. For − curvature,

the wavefunctions are curved towards the x-axis.

Therefore, we can conclude that for regions outside the well, the solutions behave much like

exponentials and within the well, the behave like superpositions **of** sine and cosine functions.

Thus, we adopt the asymptotic solution that

and

φ(x) ≈ exp(+αx)for x < a as x → −∞ (5.5)

φ(x) ≈ exp(−αx)for x > a as x → +∞ (5.6)

Finally, in the well region, φ(x) oscillates about the x-axis. We can try to obtain a more complete

solution by combining the solutions that we know. To do so, we must find solutions which are

both continuous functions **of** x and have continuous first derivatives **of** x.

Say we pick an arbitrary energy, E, and seek a solution at this energy. Define the righthand

part **of** the solution to within a multiplicative factor, then the left hand solution is then a

complicated function **of** the exact potential curve and can be written as

φIII(x) = B(E)e +ρx + B ′ (E)e −ρx

where B(E) and B ′ (E) are both real functions **of** E and depend upon the potential function.

Since the solutions must be L 2 , the only appropriate bound states are those for which B(E) = 0.

Any other value **of** B(E) leads to diverging solutions.

Thus we make the following observations concerning bound states:

1. They have negative energy.

2. They vanish exponentially outside the potential well and oscillate within.

(5.7)

3. They form a discrete spectrum as the result **of** the boundary conditions imposed by the

potential.

111

5.2 The Variational Principle

Often the interaction potential is so complicated that an exact solution is not possible. This

is **of**ten the case in molecular problems in which the potential energy surface is a complicated

multidimensional function which we know only at a few points connected by some interpolation

function. We can, however, make some approximations. The method, we shall use is the

“Variational method”.

5.2.1 Variational Calculus

The basic principle **of** the variational method lies at the heart **of** most physical principles. The

idea is that we represent a quantity as a stationary integral

J =

� x2

x1

f(y, yx, x)dx (5.8)

where f(y, yx, x) is some known function which depened upon three variables, which are also

functions **of** x, y(x), yx = dy/dx, and x itself. The dependency **of** y on x uis generally unknown.

This means that while we have fixed the end-points **of** the integral, the path that we actually

take between the endpoints is not known.

Picking different paths leads to different values **of** J. However ever certain paths will minimize,

maximize, or find the saddle points **of** J. For most cases **of** physical interest, its the extrema

that we are interested. Lets say that there is one path, yo(t) which minimizes J (See Fig. 5.1).

If we distort that path slightly, we get another path y(x) which is not too unlike yo(x) and we

will write it as y(x) = yo(x) + η(x) where η(x1) = η(x2) = 0 so that the two paths meet at the

terminal points. If η(x) differs from 0 only over a small region, we can write the new path as

and the variation from the minima as

y(x, α) = yo(x) + αη(x)

δy = y(x, α) − yo(x, 0) = αη(x).

Since yo is the path which minimizes J, and y(x, α) is some other path, then J is also a

function **of** α.

� x2

J(α) = f(y(x, α), y ′ (x, α), x)dx

x1

and will be minimized when � �

∂J

= 0

∂α α=0

Because J depends upon α, we can examine the α dependence **of** the integral

Since

� � � �

∂J

x2 ∂f ∂y ∂f

=

+

∂α

x1 ∂y ∂α ∂y α=0

′

∂y ′

�

dx

∂α

∂y

∂α

= η(x)

112

and

∂y ′

∂α

= ∂η

∂x

we have � � � �

∂J

x2 ∂f ∂f

= η(x) +

∂α

x1 ∂y ∂y α=0

′

�

∂η

dx.

∂x

Now, we need to integrate the second term by parts to get η as a common factor. Remember

integration by parts? �

�

udv = vu − vdu

From this

� x2

x1

�

∂f

∂y ′

�

∂η

dx = η(x)

∂x

∂f

�

�

�

�

∂x �

x2

x1

−

� x2

x1

η(x) d ∂f

dx

dx ∂y ′

The boundaty term vanishes since η vanishes at the end points. So putting it all together and

setting it equal to zero:

� �

x2 ∂f d ∂f

η(x) −

∂y dx ∂y ′

�

dx. = 0

x1

We’re not done yet, since we still have to evaluate this. Notice that α has disappeared from

the expression. In effect, we can take an arbitrary variation and still find the desired path tha

minimizes J. Since η(x) is arbitrary subject to the boundary conditions, we can make it have the

same sign as the remaining part **of** the integrand so that the integrand is always non-negative.

Thus, the only way for the integral to vanish is if the bracketed term is zero everywhere.

�

∂f d ∂f

−

∂y dx ∂y ′

�

= 0 (5.9)

This is known as the Euler equation and it has an enormous number **of** applications. Perhaps

the simplest is the pro**of** that the shortest distance between two points is a straight line (or

on a curved space, a geodesic). The straightline distance between two points on the xy plane

is s = √ x 2 + y 2 and the differential element **of** distance is ds =

Thus, we can write a distance along some line in the xy plane as

J =

� x2y2

x1y1

ds =

� x2y2

x1y1

�

1 + y 2 xdx.

�

(dx) 2 + (dy) 2 = �

1 + y 2 xdx.

If we knew y(x) then J would be the arclength or path-length along the function y(x) between

two points. Sort **of** like, how many steps you would take along a trail between two points. The

trail may be curvy or straight and there is certainly a single trail which is the shortest. So,

setting

f(y, yx, x) =

and substituting it into the Euler equation one gets

d ∂f

dx ∂yx

�

1 + y 2 x

= − d 1

�

dx 1 + y2 x

= 0. (5.10)

113

So, the only way for this to be true is if

1

�

1 + y2 = constant. (5.11)

x

Solving for yx produces a second constant: yx = a, which immediatly yields that y(x) = ax + b.

In other words, its a straight line! Not too surprising.

An important application **of** this principle is when the integrand f is the classical Lagrangian

for a mechanical system. The Lagrangian is related to Hamiltonian and is defined as the difference

between the kinetic and potential energy.

L = T − V (5.12)

where as H is the sum **of** T +V . Rather than taking x as the independent variable, we take time,

t, and position and velocity oa a particle as the dependent variables. The statement **of** δJ = 0

is a mathematical statement **of** Hamilton’s principle **of** least action

� t2

δ L(x, ˙x, t)dt = 0. (5.13)

t1

In essence, Hamilton’s principle asserts that the motion **of** the system from one point to another

is along a path which minimizes the integral **of** the Lagrangian. The equations **of** motion for that

path come from the Euler-Lagrange equations,

So if we write the Lagrangian as

d ∂L ∂L

−

dt ∂ ˙x ∂x

and substitute this into the Euler-Lagarange equation, we get

which is Newton’s law **of** motion: F = ma.

= 0. (5.14)

L = 1

2 m ˙x2 − V (x) (5.15)

m¨x = − ∂V

∂x

5.2.2 Constraints and Lagrange Multipliers

(5.16)

Before we can apply this principle to a quantum mechanical problem, we need to ask our selves

what happens if there is a constraint on the system which exclues certain values or paths so that

not all the η’s may be varied arbitrarily? Typically, we can write the constraint as

φi(y, x) = 0 (5.17)

For example, for a bead on a wire we need to constrain the path to always lie on the wire or for

a pendulum, the path must lie on in a hemisphere defined by the length **of** the pendulum from

114

the pivot point. In any case, the general proceedure is to introduce another function, λi(x) and

integrate

so that

� x2

x1

λi(x)φi(y, x)dx = 0 (5.18)

� x2

δ λi(x)φi(y, x)dx = 0 (5.19)

x1

as well. In fact it turns out that the λi(x) can be even be taken to be a constant, λi for this

whole proceedure to work.

Regardless **of** the case, we can always write the new stationary integral as

�

δ

(f(y, yx, x) + �

λiφi(y, x))dx = 0. (5.20)

i

The multiplying constants are called Lagrange Mulipliers. In your statistical mechanics course,

these will occur when you minimize various thermodynamic functions subject to the various

extensive constraints, such as total number **of** particles in the system, the average energy or

temperature, and so on.

In a sence, we have redefined the original function or Lagrangian to incorporate the constraint

into the dynamics. So, in the presence **of** a constraint, the Euler-Lagrange equations become

d ∂L ∂L

−

dt ∂ ˙x ∂x

= �

i

∂φi

∂x λi

(5.21)

where the term on the right hand side **of** the equation represents a force due to the constraint.

The next issue is that we still need to be able to determine the λi Lagrange multipliers.

115

Figure 5.1: Variational paths between endpoints. The thick line is the stationary path, yo(x)

and the dashed blue curves are variations y(x, α) = yo(x) + αη(x).

fHxL

2

1

-1.5 -1 -0.5 0.5 1 1.5 x

-1

-2

116

5.2.3 Variational method applied to Schrödinger equation

The goal **of** all this is to develop a procedure for computing the ground state **of** some quantum

mechanical system. What this means is that we want to minimize the energy **of** the system

with respect to arbitrary variations in the state function subject to the constraint that the state

function is normalized (i.e. number **of** particles remains fixed). This means we want to construct

the variation:

δ〈ψ|H|ψ〉 = 0 (5.22)

with the constraint 〈ψ|ψ〉 = 0.

In the coordinate representation, the integral involves taking the expectation value **of** the

kinetic energy operator...which is a second derivative operator. That form is not too convenient

for our purposes since it will allow us to write Eq.5.22 in a form suitable for the Euler-Lagrange

equations. But, we can integrate by parts!

�

ψ ∗ ∂2 �

ψ ∂ψ �

�

dx = ψ∗ �

∂x2 ∂x � −

� � ∂ψ∗ � � �

∂ψ

dx (5.23)

∂x ∂x

Assuming that the wavefunction vanishes at the limits **of** the integration, the surface term vanishes

leaving only the second term. We can now write the energy expectation value in terms

**of** two dependent variables, ∇ψ and ψ. OK, they’re functions, but we can still treat them as

dependent variables just like we treated the y(x) ′ s above.

�

E =

� ¯h 2

2m (∇ψ∗ )(∇ψ) + V ψ ∗ �

ψ dx (5.24)

Adding on the constraint and defining the Lagrangian as

L =

� ¯h 2

2m (∇ψ∗ )(∇ψ) + V ψ ∗ ψ

we can substitute this into the Euler-Lagrange equations

This produces the result

�

− λψ ∗ ψ, (5.25)

∂L ∂L

− ∇∂x = 0. (5.26)

∂ψ∗ ∂(∇ψ∗ (V − λ)ψ = ¯h2

2m ∇2 ψ, (5.27)

which we immediately recognize as the Schrödinger equation.

While this may be a rather academic result, it gives us the key to recognize that we can

make an expansion **of** ψ in an arbitrary basis and take variations with respect to the coeffients

**of** that basis to find the lowest energy state. This is the basis **of** a number **of** powerful numerical

methods used solve the Schrödinger equation for extremely complicated systems.

117

5.2.4 Variational theorems: Rayleigh-Ritz Technique

We now discuss two important theorems:

Theorem 5.1 The expectation value **of** the Hamiltonian is stationary in the neighborhood **of** its

eigenstates.

To demonstrate this, let |ψ〉 be a state in which we compute the expectation value **of** H. Also,

let’s modify the state just a bit and write

Expectation values are computed as

|ψ〉 → |ψ〉 + |δψ〉. (5.28)

〈H〉 = 〈ψ|H|ψ〉

〈ψ|ψ〉

(where we assume arb. normalization). In other words

Now, insert the variation

or

(5.29)

〈ψ|ψ〉〈H〉 = 〈ψ|H|ψ〉 (5.30)

〈ψ|ψ〉δ〈H〉 + 〈δψ|ψ〉〈H〉 + 〈ψ|δψ〉〈H〉 = 〈δψ|H|ψ〉 + 〈ψ|H|δψ〉 (5.31)

〈ψ|ψ〉δ〈H〉 = 〈δψ|H − 〈H〉|ψ〉 − 〈ψ|H − 〈H〉|δψ〉 (5.32)

If the expectation value is to be stationary, then δ〈H〉 = 0. Thus the RHS must vanish for an

arbitrary variation in the wavefunction. Let’s pick

Thus,

|δψ〉 = ɛ|ψ〉. (5.33)

(H − 〈H〉)|ψ〉 = 0 (5.34)

That is to say that |ψ〉 is an eigenstate **of** H. Thus proving the theorem.

The second theorem goes:

Theorem 5.2 The Expectation value **of** the Hamiltonian in an arbitrary state is greater than or

equal to the ground-state energy.

The pro**of** goes as this: Assume that H has a discrete spectrum **of** states (which we demonstrated

that it must) such that

H|n〉 = En|n〉 (5.35)

118

Thus, we can expand any state |ψ〉 as

Consequently

and

Thus, (assuming that |ψ〉 is normalized)

〈H〉 = �

En|cn| 2 ≥ �

Eo|cn| 2 ≥ Eo

|ψ〉 = �

cn|n〉. (5.36)

n

〈ψ|ψ〉 = �

|cn| 2 , (5.37)

n

〈ψ|H|ψ〉 = �

|cn| 2 En. (5.38)

n

n

n

(5.39)

quid est demonstrato.

Using these two theorems, we can estimate the ground state energy and wavefunctions for a

variery **of** systems. Let’s first look at the Harmonic Oscillator.

Exercise 5.1 Use the variational principle to estimate the ground-state energy **of** a particle in

the potential

�

Cx x > 0

V (x) =

(5.40)

+∞ x ≤ 0

Use xe −ax as the trial function.

5.2.5 Variational solution **of** harmonic oscillator ground State

The Schrödinger Equation for the Harmonic Osc. (HO) is

Take as a trial function,

�

− ¯h2 ∂

2m

2

∂x

2 + k2

2 x2

�

φ(x) − Eφ(x) = 0 (5.41)

φ(x) = exp(−λx 2 ) (5.42)

where λ is a positive, non-zero constant to be determined. The variational principle states that

the energy reaches a minimum

∂〈H〉

∂λ

when φ(x) is the ground state solution. Let us first derive 〈H〉(λ).

〈H〉(λ) = 〈φ|H|φ〉

〈φ|φ〉

= 0. (5.43)

119

(5.44)

To evaluate this, we break the problem into a series **of** integrals:

and

Putting it all together:

〈φ|p 2 � ∞

|φ〉 =

� ∞

〈φ|φ〉 = dx|φ(x)| 2 =

−∞

� π

2λ

(5.45)

dxφ

−∞

′′ (x)φ(x) = −2λ〈φ|φ〉 + 4λ 2 〈φ|x 2 |φ〉 (5.46)

< φ|x 2 � ∞

|φ〉 =

Taking the derivative with respect to λ:

Thus,

−∞

dxx 2 |φ(x)| 2 = 1

〈φ|φ〉. (5.47)

4λ

〈φ|H|φ〉

〈φ|φ〉 =

�

− ¯h2

� �

�

2 1

−2λ + 4λ +

2m

4λ

k 1

2 4λ

〈φ|H|φ〉

〈φ|φ〉 =

� �

2

¯h

λ +

2m

k

8λ

∂〈H〉

∂λ

Since only positive values **of** λ are allowed.

= ¯h2

2m

λ = ±

λ =

(5.48)

(5.49)

k

− = 0 (5.50)

8λ2 √ mk

2¯h

√ mk

2¯h

Using this we can calculate the ground state energy by substituting λ back into 〈H〉(λ).

〈H〉(λ) =

Now, define the angular frequency: ω =

√ �

2

mk ¯h

2¯h 2m

�

k/m.

k 4¯h

+

8

2

�

=

mk

¯h

�

k

2 m

(5.51)

(5.52)

(5.53)

〈H〉(λ) = ¯h

ω (5.54)

2

which ( as we can easily prove) is the ground state energy **of** the harmonic oscillator.

Furthermore, we can write the HO ground state wavefunction as

120

φo(x) =

φo(x) =

1

φo(x) = � φ(x) (5.55)

〈φ|φ〉

� �1/4 � √

2λ

mk

exp −

π

2¯h x2

�

�√ mk

¯hπ

�1/4 � √

mk

exp −

2¯h x2

�

(5.56)

(5.57)

To compute the “error” in our estimate, let’s substitute the variational solution back into the

Schrodinger equation:

�

− ¯h2 ∂

2m

2

∂x

2 + k2

2 x2

�

φo(x) = − ¯h2

2m φ′′

o(x) + k2

2 φo(x) (5.58)

− ¯h2

2m φ′′ o(x) + k2

2 φo(x) = − ¯h2

� √

2 kmx − ¯h km

2m ¯h 2

�

φo(x) + k

2 x2φo(x) (5.59)

− ¯h2

2m φ′′ o(x) + k2

2 φo(x) = ¯h

�

k

2 m φo(x) (5.60)

Thus, φo(x) is in fact the correct ground state wavefunction for this system. If it were not

the correct function, we could re-introduce the solution as a new trial function, re-compute the

energy, etc... and iterate through until we either find a solution, or run out **of** patience! (Usually

it’s the latter than the former.)

5.3 The Harmonic Oscillator

Now that we have the HO ground state and the HO ground state energy, let us derive the whole

HO energy spectrum. To do so, we introduce “dimensionless” quantities: X and P related to

the physical position and momentum by

X = ( mω

2¯h )1/2 x (5.61)

1

P = (

2¯hmω )1/2p (5.62)

This will save us from carrying around a bunch **of** coefficients. In these units, the HO Hamiltonian

is

H = ¯hω(P 2 + X 2 ). (5.63)

121

The X and P obey the canonical comutational relation:

We can also write the following:

Thus, I can construct the commutator:

[X, P ] = 1 i

[x, p] =

2¯h 2

(5.64)

(X + iP )(X − iP ) = X 2 + P 2 + 1/2 (5.65)

(X − iP )(X + iP ) = X 2 + P 2 − 1/2. (5.66)

[(X + iP ), (X − iP )] = (X + iP )(X − iP ) − (X − iP )(X + iP )

Let’s define the following two operators:

Therefore, the a and a † commute as

= 1/2 + 1/2

Let’s write H in terms **of** the a and a † operators:

or in terms **of** the a and a † operators:

= 1 (5.67)

a = (X + iP ) (5.68)

a † = (X + iP ) † = (X − iP ). (5.69)

[a, a † ] = 1 (5.70)

H = ¯hω(X 2 + P 2 ) = ¯hω(X − iP )(X + iP ) + ¯hω/2 (5.71)

H = ¯hω(a † a + 1/2) (5.72)

Now, consider that |φn〉 is the nth eigenstate **of** H. Thus, we write:

¯hω(a † a + 1/2)|φn〉 = En|φn〉 (5.73)

What happens when I multiply the whole equation by a? Thus, we write:

Now, since aa † − a † a = 1,

a¯hω(a † a + 1/2)|φn〉 = aEn|φn〉 (5.74)

¯hω(aa † + 1/2)(a|φn〉) = En(a|φn〉) (5.75)

¯hω(a † a + 1/2 − 1)(a|φn〉) = En(a|φn〉) (5.76)

In other words, a|φn〉 is an eigenstate **of** H with energy E = En − ¯hω.

122

Since

What happends if I do the same procedure, this time using a † ? Thus, we write:

we have

we can write

Thus,

or,

a † ¯hω(a † a + 1/2)|φn〉 = a † En|φn〉 (5.77)

[a, a † ] = aa † − a † a (5.78)

a † a = aa † − 1 (5.79)

a † a † a = a † (aa † − 1) (5.80)

= (a † a − 1)a † . (5.81)

a † ¯hω(a † a + 1/2)|φn〉 = ¯hω((a † a − 1 + 1/2)a † )|φn〉 (5.82)

¯hω(a † a − 1/2)(a † |φn〉) = En(a † |φn〉). (5.83)

Thus, a † |φn〉 is an eigenstate **of** H with energy E = En + ¯hω.

Since a † and a act on harmonic oscillator eigenstates to give eigenstates with one more or one

less ¯hω quanta **of** energy, these are termed “creation” and “annihilation” operators since they

act to create additional quata **of** excitation or decrease the number **of** quanta **of** excitation in

the system. Using these operators, we can effectively “ladder” our way up the energy scale and

determine any eigenstate one we know just one.

Well, we know the ground state solution. That we got via the variational calculation. What

happens when I apply a † to the φo(x) we derived above? In coordinate form:

(X − iP ) φo(x) =

X acting on φo is:

iP acting on φo is

=

� �mω

2¯h

� �mω

2¯h

Xφo(x) =

�1/2 � �1/2

�

1 ∂

x +

φo(x)

2mω¯h ∂x

�1/2 � �1/2 1 ∂

x +

2mω¯h ∂x

� �mω

¯hπ

(5.84)

�1/4 mω

(−x2 e 2¯h ) (5.85)

� mω

2¯h xφo(x) (5.86)

�

iP φo(x) = −¯h

1 ∂

mω2¯h ∂x φo(x) (5.87)

123

After cleaning things up:

iP φo(x) =

�

mω

−x

2¯h φo(x) (5.88)

= −Xφo(x) (5.89)

(X − iP ) φo(x) =

=

2Xφo(x)

�

mω

2

2¯h

(5.90)

xφo(x) (5.91)

=

=

2Xφo(x)

�

mω

2

2¯h

(5.92)

x

� �

mω 1/4 � �

2 mω

exp −x

2¯h

2¯h

(5.93)

5.3.1 Harmonic Oscillators and Nuclear Vibrations

We introduced one **of** the most important applications **of** quantum mechanics...the solution **of**

the Schrödinger equation for harmonic systems. These are systems in which the amplitude **of**

motion is either small enough so that the physical potential energy operator can be expanded

about its minimum to second order in the displacement from the minima. When we do so, the

Hamiltonian can be written in the form

H = ¯hω(P 2 + X 2 ) (5.94)

where P and X are dimensionless operators related to the physical momentum and position

operators via

�

mω

X = x (5.95)

2¯h

and

�

P =

1

p.

2¯hmω

(5.96)

We also used the variational method to deduce the ground state wavefunction and demonstrated

that the spectrum **of** H is a series **of** levels separated by ¯hω and that the ground-state energy is

¯hω/2 above the energy minimum **of** the potential.

We also defined a new set **of** operators by taking linear combinations **of** the X and P .

a = X + iP (5.97)

a † = X − iP. (5.98)

We also showed that the commutation relation for these operators is

[a, a † ] = 1. (5.99)

These operators are non-hermitian operators, and hence, do not correspond to a physical observable.

However, we demonstrated that when a acts on a eigenstate **of** H, it produces another

124

eigenstate withe energy En − ¯hω. Also, a † acting on an eigenstate **of** H produces another eigenstate

with energy En + ¯hω. Thus,we called a the destruction or annihilation operator since it

removes a quanta **of** excitation from the system and a † the creation operator since it adds a

quanta **of** excitation to the system. We also wrote H using these operators as

H = ¯hω(a † a + 1

) (5.100)

2

Finally, ω is the angular frequency **of** the classical harmonic motion, as obtained via Hooke’s

law:

¨x = − k

x. (5.101)

m

Solving this produces

and

x(t) = xo sin(ωt + φ) (5.102)

p(t) = po cos(ωt + φ). (5.103)

Thus, the classical motion in the x, p phase space traces out the circumference **of** a circle every

1/ω regardless **of** the initial amplitude.

The great advantage **of** using the a, and a † operators is that they we can replace a differential

equation with an algebraic equation. Furthermore, since we can represent any Hermitian operator

acting on the HO states as a combination **of** the creation/annihilation operators, we can replace

a potentially complicated series **of** differentiations, integrations, etc... with simple algebraic

manipulations. We just have to remember a few simple rules regarding the commutation **of** the

two operators. Two operators which we may want to construct are:

and

• position operator: � �1/2 2¯h (a mω

† + a)

• momentum operator: i � �1/2 ¯hmω (a 2

† − a).

Another important operator is

N = a † a. (5.104)

H = ¯hω(N + 1/2). (5.105)

Since [H, N] = 0, eigenvalues **of** N are “good quantum numbers” and N is a constant **of** the

motion. Also, since

then if

H|φn〉 = En|φn〉 = ¯hω(N + 1/2)|φn〉 (5.106)

N|φn〉 = n|φn〉, (5.107)

then n must be an integer n = 0, 1, 2, · · · corresponding to the number **of** quanta **of** excitation in

the state. This gets the name “Number Operator”.

Some useful relations (that you should prove )

125

1. [N, a] = [a † a, a] = −a

2. [N, a † ] = [a † a, a † ] = a †

To summarize, we have the following relations using the a and a † operators:

1. a|φn〉 = √ n|φn−1〉

2. a † |φn〉 = √ n + 1|φn+1〉

3. 〈φn|a = √ n + 1〈φn+1 = (a|φn〉) †

4. 〈φn|a † = √ n + 1〈φn−1|

5. N|φn〉 = n|φn〉

6. 〈φn|N = n〈φn|

Using the second **of** these relations we can write

which can be iterated back to the ground state to produce

This is the “generating relation” for the eigenstates.

Now, let’s look at x and p acting on |φn〉.

Also,

|φn+1〉 = a†

√ n + 1 |φn〉 (5.108)

|φn〉 = (a† ) n

√ n! |φo〉 (5.109)

�

x|φn〉 =

¯h

2mω (a† + a)|φn〉 (5.110)

�

¯h

=

2mω (√n + 1|φn+1〉 + √ n|φn−1〉) (5.111)

= i

�

p|φn〉 = i

�

m¯hω

2 (a† − a)|φn〉 (5.112)

m¯hω

2 (√ n + 1|φn+1〉 − √ n|φn−1〉) (5.113)

Thus,the matrix elements **of** x and p in the HO basis are:

�

¯h �√

〈φm|x|φn〉 = n + 1δm,n+1 +

2mω

√ �

nδm,n−1

126

(5.114)

〈φm|p|φn〉 = i

�

mω¯h

2

�√ n + 1δm,n+1 − √ nδm,n−1

The harmonic oscillator wavefunctions can be obtained by solving the equation:

�

(5.115)

〈x|a|φo〉 = (X + iP )φo(x) = 0 (5.116)

� mω

The solution **of** this first order differential equation is easy:

¯h

�

∂

x + φo(x) = 0 (5.117)

∂x

φo(x) = c exp(− mω

2 x2 ) (5.118)

where c is a constant **of** integration which we can obtain via normalization:

Doing the integration produces:

�

φo(x) =

dx|φo(x)| 2 = 1 (5.119)

� �

mω 1/4

mω

−

e 2¯h

¯hπ

x2

127

(5.120)

Figure 5.2: Hermite Polynomials, Hn up to n = 3.

HnHxL

10

7.5

5

2.5

-3 -2 -1 1 2 3 x

-2.5

-5

-7.5

-10

128

Since we know that a † acting on |φo〉 gives the next eigenstate, we can write

φ1(x) =

� mω

¯h

�

∂

x − φo(x) (5.121)

∂x

Finally, using the generating relation, we can write

φn(x) = 1

� �n mω ∂

√ x − φo(x). (5.122)

n! ¯h ∂x

Lastly, we have the “recursion relations” which generates the next solution one step higher or

lower in energy given any other solution.

� �

1 mω ∂

φn+1(x) = √ x − φn(x) (5.123)

n + 1 ¯h ∂x

and

φn−1(x) = 1

� �

mω ∂

√ x + φn(x). (5.124)

n ¯h ∂x

These are the recursion relationships for a class **of** polynomials called Hermite polynomials, after

the 19th French mathematician who studied such functions. These are also termed “Gauss-

Hermite” and form a set **of** orthogonal polynomials. The first few Hermite Polynomials, Hn(x)

are {1, 2 x, −2 + 4 x 2 , −12 x + 8 x 3 , 12 − 48 x 2 + 16 x 4 } for n = 0 to 4. Some **of** these are plotted

in Fig. 5.2

The functions themselves are defined by the generating function

g(x, t) = e −t2 ∞�

+2tx

=

n=0

Hn(x) tn

. (5.125)

n!

Differentiating the generating function n times and setting t = 0 produces the nth Hermite

polynomial

Hn(x) = dn

�

�

�

g(x, t) �

dtn � = (−1)n dn x2

e e−x2

(5.126)

dxn Another useful relation is the Fourier transform relation:

�

1 ∞

√

2π −∞

e itx e −x2 /2 Hn(x)dx = −i n e −t2 /2 Hn(t) (5.127)

which is useful in generating the momentum space representation **of** the harmonic oscillator

functions. Also, from the generating function, we can arrive at the recurrence relation:

and

Hn+1 = 2xHn − 2nHn−1

(5.128)

H ′ n(x) = 2nHn−1(x). (5.129)

129

Consequently, the hermite polynomials are solutions **of** the second-order differental equation:

H ′′

n − 2xH ′ n + 2nHn = 0 (5.130)

which is not self-adjoint! To put this into self-adjoint form, we multiply by the weighting function

w = e−x2, which leads to the orthogonality integral

� ∞

Hn(x)Hm(x)e −x2

dx = δnm. (5.131)

−∞

For the harmonic oscillator functions, we absorb the weighting function into the wavefunction

itself

ψn(x) = e −x2 /2 Hn(x).

When we substitute this function into the differential equation for Hn we get

ψ ′′

n + (2n + 1 − x 2 )ψn = 0. (5.132)

To normalize the functions, we first multipy g by itself and then multiply by w

e −x2

e −s2 +2sx −t

e 2 �

+2tx

=

mn

e −x2

Hn(x)Hm(x) smtn n!m!

(5.133)

When we integrate over −∞ to ∞ the cross terms drop out by orthogonality and we are left

with

� (st)

n=0

n � ∞

e

n!n! −∞

−x2

H 2 n(x)dx =

� ∞

e

−∞

−x2−s2 +2sx−t2 =

+2xt

dx

� ∞

e

−∞

−(x−s−t)2

dx

= π 1/2 e 2st = � 2

n=0

n (st) n

.

n!

(5.134)

Equating like powers **of** st we obtain,

� ∞

e −x2

H 2 n(x)dx = 2 n π 1/2 n!. (5.135)

When we apply this technology to the SHO, the solutions are

where z = αx and

A few gratuitous solutions:

−∞

ψn(z) = 2 −n/2 π −1/4 (n!) −1/2 e −z2

Hn(z) (5.136)

φ1(x) =

φ2(x) =

� 4

π

Fig. 5.3 shows the first 4 **of** these functions.

α 2 = mω

¯h .

� � �

mω 3 1/4

x exp(−

¯h

1

2 mωx2 ) (5.137)

� �

mω 1/4 �

2

4π¯h

mω

¯h x2 �

− 1 exp(− 1

2 mωx2 ) (5.138)

130

5.3.2 Classical interpretation.

In Fig. 5.3 are a few **of** the lowest energy states for the harmonic oscillator. Notice that as the

quantum number increases the amplitude **of** the wavefunction is pushed more and more towards

larger values **of** ±x. This becomes more pronounced when we look at the actual probability

distribution functions, |ψn(x)| 2 | for the same 4 states as shown in Fig 5.4.

Here, in blue are the actual quantum distributions for the ground-state through n = 3. In

gray are the classical probability distrubutions for the corresponding energies. The gray curves

tell us the probabilty per unit length **of** finding the classical particle at some point x and any

point in time. This is inversely proportional to how long a particle spends at a given point...i.e.

Pc(x) ∝ 1/v(x). Since E = mv 2 /2 + V (x),

and

For the Harmonic Oscillator:

v(x) =

�

2(E − V (x))/m

�

m

P (x) ∝

2(E − V (x))

�

m

Pn(x) ∝

2(¯hω(n + 1/2) − kx2 /2) .

Notice that the denominator goes to zero at the classical turning points, in other words,

the particle comes to a dead-stop at the turning point and consequently we have the greatest

likelyhood **of** finding the particle in these regions. Likewise in the quantum case, as we increase

the quantum number, the quantum distrubution function becomes more and more like its classical

counterpart. This is shown in the last four frames **of** Fig. 5.4 where we have the same plots as in

Fig. 5.4, except we look at much higher quantum numbers. For the last case, where n = 19 the

classical and quantum distributions are nearly identical. This is an example **of** the correspondence

principle. As the quantum number increases, we expect the quantum system to look more and

more like its classical counter part.

131

1

0.8

0.6

0.4

0.2

-4 -2 2 4

2

1

-4 -2 2 4

-1

-2

Figure 5.3: Harmonic oscillator functions for n = 0 to 3

1

0.5

-4 -2 2 4

-0.5

-1

4

2

-4 -2 2 4

-2

-4

132

Figure 5.4: **Quantum** and Classical Probability Distribution Functions for Harmonic Oscillator

for n = 0, 1, 2, 3, 4, 5, 9, 14, 19

2.5

2

1.5

1

0.5

-4 -2 2 4

1.5

1.25

1

0.75

0.5

0.25

-4 -2 2 4

0.8

0.6

0.4

0.2

-7.5 -5 -2.5 2.5 5 7.5

0.6

0.5

0.4

0.3

0.2

0.1

-7.5 -5 -2.5 2.5 5 7.5

1.75

1.5

1.25

1

0.75

0.5

0.25

-4 -2 2 4

1.4

1.2

1

0.8

0.6

0.4

0.2

-4 -2 2 4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

-7.5 -5 -2.5 2.5 5 7.5

0.6

0.5

0.4

0.3

0.2

0.1

-7.5 -5 -2.5 2.5 5 7.5

133

5.3.3 Molecular Vibrations

The fully quantum mechanical treatment **of** both the electronic and nuclear dynamics **of** even a

diatomic molecule is a complicated affair. The reason for this is that we are forced to find the

stationary states for a potentially large number **of** particles–all **of** which interact, constrained

by a number **of** symmetry relations (such as the fact that no two electrons can be in the same

state at the same time.) In general, the exact solution **of** a many-body problem (such as this)

is impossible. (In fact, believe it it is rigorously impossible for even three classically interacting

particles..although many have tried. ) However, the mass **of** the electron is on the order **of** 10 3 to

10 4 times smaller than the mass **of** a typical nuclei. Thus, the typical velocities **of** the electrons

is much larger than the typical nuclear velocities. We can then assume that the electronic cloud

surrounding the nuclei will respond instantaneously to small and slow changes to the nuclear

positions. Thus, to a very good approximation, we can separate the nuclear motion from the

electonic motion. This separation **of** the nuclear and electronic motion is called the Born-

Oppenheimer Approximation or the Adiabatic Approximation. This approximation is

one **of** the MOST important concepts in chemical physics and is covered in more detail in Section

8.4.1.

Fundimental notion is that the nuclear motion **of** a molecule occurs in the average field **of**

the electrons. In other words, the electronic charge distribution acts as an extremely complex

multi-dimensional potential energy surface which governs the motion and dynamics **of** the atoms

in a molecule. Consequently, since chemistry is the science **of** chemical structure, changes, and

dynamics, nearly all chemical reactions can be described in terms **of** nuclear motion on one (or

more) potential energy surface. In Fig. ?? is the London-Eyring-Polanyi-Sato (LEPS) [1]surface

for the F + H2 → HF + H reaction using the Mukerman V set **of** parameters.[3] The LEPS

surface is an empirical potential energy surface based upon the London-Heitler valance bond

theory. Highly accurate potential functions are typically obtained by performing high level ab

initio electronic structure calculations sampling over numerous configurations **of** the molecule.[2]

For diatomic molecules, the nuclear stretching potential can be approximated as a Morse

potential curve

V (r) = De(1 − e −α(r−req ) 2 − De

(5.139)

where De is the dissociation energy, α sets the range **of** the potential, and req is the equilibrium

bond length. The Morse potential for HF is shown in Fig. 5.6 and is parameterized by De =

591.1kcal/mol, α = 2.2189˚A −1 , and req = 0.917˚A.

Close to the very bottom **of** the potential well, where r − re is small, the potential is nearly

harmonic and we can replace the nuclear SE with the HO equation by simply writing that the

angular frequancy is

�

V

ω =

′′ (re)

(5.140)

m

So, measuring the vibrational spectrum **of** the well will give us the curvature **of** the well since

(En − Em)/¯h is always an integer multiple **of** ω for harmonic systems. The red curve in Fig. 5.6

is a parabolic approximation for the bottom **of** the well.

V (r) = De(−1 + α 2 (r − re) 2 /2 + . . .) (5.141)

134

Figure 5.5: London-Eyring-Polanyi-Sato (LEPS) empirical potential for the F + H2 → F H + H

chemical reaction

3.5

3

2.5

2

1.5

1

0.5

rHH

4

0.5 1 1.5 2 2.5 3 3.5 4

135

rFH

V HkCalêmolL

600

400

200

-200

-400

-600

Figure 5.6: Morse well and harmonic approximation for HF

1 2 3 4

Clearly, Deα2 �

is the force constant. So the harmonic frequency for the well is ω = k/µ, where

µ is the reduced mass, µ = m1m2/(m1 + m2) and one would expect that the vibrational energy

levels would be evenly spaced according to a harmonic progression. Deviations from this are due

to anharmonic effects introduced by the inclusion **of** higher order terms in the Taylor expansion

**of** the well. As one might expect, the harmonic expansion provides a descent estimate **of** the

potential energy surface close to the equilibrium geometry.

5.4 Numerical Solution **of** the Schrödinger Equation

5.4.1 Numerov Method

Clearly, finding bound state soutions for the Schrödinger equation is an important task. Unfortunately,

we can only solve a few systems exactly. For the vast majority **of** system which

we cannot handle exactly, we need to turn to approximate means to finde solutions. In later

chapters, we will examine variational methods and perturbative methods. Here, we will look at

a very simple scheme based upon propagating a trial solution at a given point. This methods is

called the ”Numerov” approach after the Russian astronomer who developed the method. It can

be used to solve any differential equation **of** the form:

f ′′ (r) = f(r)u(r) (5.142)

where u(r) is simply a function **of** r and f ′′ is the second derivative **of** f(r) with respect to r and

which is the solution we are looking for. For the Schrödinger equation we would write:

ψ ′′ = 2m

2 (V (r) − E)ψ (5.143)

¯h

136

rFH

Figure 5.7: Model potential for proton tunneling.

V Hcm

6000

-1L 4000

2000

-1 -0.5 0.5 1

-2000

-4000

x HbohrL

The basic proceedure is to expand the second derivative as a finite difference so that if we know

the solution at one point, xn, we can get the solution at some small point a distance h away,

xn+1 = xn + h.

f[n + 1] = f[n] + hf ′ [n] + h2

2! f ′′ [n] + h3

3! f ′′′ [n] + h4

4! f (4) [n] . . . (5.144)

f[n − 1] = f[n] − hf ′ [n] + h2

2! f ′′ [n] − h3

3! f ′′′ [n] + h4

4! f (4) [n] (5.145)

If we combine these two equations and solve for f[n + 1] we get a result:

f[n + 1] = −f[n − 1] + 2f[n] + f ′′ [n]h 2 + h4

12 f (4) [n] + O[h 6 ] (5.146)

Since f is a solution to the Schrödinger equation, f ′′ = Gf where

G = 2m

2 (V − E)

¯h

we can get the second derivative **of** f very easily. However, for the higher order terms, we have

to work a bit harder. So let’s expand f ′′

f ′′ [n + 1] = −2f ′′ [n] − f ′′ [n − 1] + s 2 f (4) [n]

and truncate at order h 6 . Now, solving for f (4) [n] and substituting f ′′ = Gf we get

f[n + 1] =

�

2f[n] − f[n − 1] + h2 (G[n − 1]f[n − 1] + 10G[n]f[n])�

12

�

1 − h2

(5.147)

G[n + 1]�

12

which is the working equation.

Here we take a case **of** proton tunneling in a double well potential. The potential in this case

is the V (x) = α(x 4 − x 2 ) function shown in Fig. 5.7. Here we have taken the parameter α = 0.1

137

1

0.5

-2 -1 1 2

-0.5

-1

-2 -1 1 2

Figure 5.8: Double well tunneling states as determined by the Numerov approach. On the left

is the approximate lowest energy (symmetric) state with no nodes and on the rigth is the next

lowest (antisymmetric) state with a single node. The fact that the wavefunctions are heading **of**f

towards infinity indicated the introduction **of** an additional node coming in from x = ∞.

and m = 1836 as the proton and use atomic units throughout. Also show in Fig. 5.7 are effective

harmonic oscillator wells for each side **of** the barrier. Notice that the harmonic approximation

is pretty crude since the harminic well tends to over estimate the steepness **of** the inner portion

and under estimate the steepness **of** the outer portions. Nonetheless, we can use the harminic

oscillator ground states in each well as starting points.

To use the Numerov method, one starts by guessing an initial energy, E, and then propagating

a trial solution to the Schrödinger equation. The Curve you obtain is in fact a solution to the

equation, but it will ususally not obey the correct boundary conditions. For bound states, the

boundary condition is that ψ must vanish exponentially outside the well. So, we initialize the

method by forcing ψ[1] to be exactly 0 and ψ[2] to be some small number. The exact values

really make no difference. If we are **of**f by a bit, the Numerov wave will diverge towards ±∞ as x

increases. As we close in on a physically acceptible solution, the Numerov solution will begin to

exhibit the correct asymptotic behavour for a while before diverging. We know we have hit upon

an eigenstate when the divergence goes from +∞ to −∞ or vice versa, signeling the presence

**of** an additional node in the wavefunction. The proceedure then is to back up in energy a bit,

change the energy step and gradually narrow in on the exact energy. In Figs. 5.8a and 5.8b are

the results **of** a Numerov search for the lowest two states in the double well potential. One at

-3946.59cm −1 and the other at -3943.75cm −1 . Notice that the lowest energy state is symmetric

about the origin and the next state is anti-symmetric about the origin. Also in both cases, the

Numerov function diverges since we are not precisely at a stationary solution **of** the Schrödinger

equation...but we are within 0.01cm −1 **of** the true eigenvalue.

The advantage **of** the Numerov method is that it is really easy to code. In fact you can

even code it in Excell. Another advantage is that for radial scattering problems, the out going

boundary conditions occur naturally, making it a method **of** choice for simple scattering problems.

In the Mathematica notebooks, I show how one can use the Numerov method to compute the

scattering phase shifts and locate resonance for atomic collisions. The disadvantage is that you

have to search by hand for the eigenvalues which can be extremely tedious.

138

3

2

1

-1

5.4.2 Numerical Diagonalization

A more general approach is based upon the variational principle (which we will discuss later)

and the use **of** matrix representations. If we express the Hamiltonian operator in matrix form in

some suitable basis, then the eigenfunctions **of** H can also be expressed as linear combinations

**of** those basis functions, subject to the constraint that the eigenfunctions be orthonormal. So,

what we do is write:

〈φn|H|φm〉 = Hnm

and

|ψj〉 = �

〈φn|ψj〉|φn〉

n

The 〈φn|ψj〉 coefficients are also elements **of** a matrix, Tnj which transforms a vector in the φ

basis to the ψ basis. Conseqently, there is a one-to-one relation between the number **of** basis

functions in the ψ basis and the basis functions in the φ basis.

If |ψn〉 is an eigenstate **of** H, then

Multiplying by 〈φm| and resolving the identity,

Thus,

or in more compact form

H|ψj〉 = Ej|ψj〉.

�

〈φm|H|φn〉〈φn|ψj〉 = 〈φm|ψj〉Ej

n

�

HmnTnj

n

= EjTmj (5.148)

�

TmjHmnTnj = Ej

mn

T † HT = E Î

(5.149)

where Î is the identity matrix. In otherwords, the T -matrix is simply the matrix which brings

H to diagonal form.

Diagonalizing a matrix by hand is very tedeous for anything beyond a 3×3 matrix. Since this

is an extremely common numerical task, there are some very powerful numerical diagonalization

routines available. Most **of** the common ones are in the Lapack package and are included as part

**of** the Mathematica kernel. So, all we need to do is to pick a basis, cast our Hamiltonian into

that basis, truncate the basis (usually determined by some energy cut-**of**f) and diagonalize away.

Usually the diagonalization part is the most time consuming. Of course you have to be prudent

in choosing your basis.

A useful set **of** basis functions are the trigonmetric forms **of** the Tchebychev polynomials. 1

These are a set **of** orthogonal functions which obey the following recurrence relation

Tn+1(x) − 2xTn(x) + Tn−1(x) = 0 (5.150)

139

1

0.75

0.5

0.25

-1 -0.5 0.5 1

-0.25

-0.5

-0.75

-1

Figure 5.9: Tchebyshev Polynomials for n = 1 − 5

Table 5.1: Tchebychev polynomials **of** the first type

To = 1

T1 = x

T2 = 2x 2 − 1

T3 = 4x 3 − 3x

T4 = 8x 4 − 8x 2 − 1

T5 = 16x 5 − 20x 3 + 5x

140

Table 5.1 lists a the first few **of** these polynomials as functions **of** x and a few **of** these are plotted

in Fig. 5.9

It is important to realize that these functions are orthogonal on a finite range and that

integrals over these functions must include a weighting function w(x) = 1/ √ 1 − x 2 . The orthogonality

relation for the Tn polynomials is

� +1

−1

⎧

⎪⎨ 0 m �= m

π

Tm(x)Tn(x)w(x)dx = m = n �= 0

⎪⎩

2

π m = n = 0

(5.151)

Arfkin’s Mathematical Methods for Physicists has a pretty complete overview **of** these special

functions as well as many others. As usual, These are encorporated into the kernel **of** Mathematica

and the Mathematica book and on-line help pages has some useful information regarding

these functions as well as a plethera **of** other functions.

From the recurrence relation it is easy to show that the Tn(x) polynomials satisfy the differential

equation:

(1 − x 2 )T ′′

n − xT ′ x + n 2 Tn = 0 (5.152)

If we make a change **of** variables from x = cos(θ) and dx = − sin θdθ, then the differential

equation reads

dTn

dθ + n2 Tn = 0 (5.153)

This is a harmonic oscillator and has solutions sin nθ and cos nθ. From the boundary conditions

we have two linearly independent solutions

and

The normalization condition then becomes:

and

� +1

−1

� +1

−1

Tn = cos nθ = cos n(arccosx)

Vn = sin nθ.

Tm(x)Tn(x)w(x)dx =

Vm(x)Vn(x)w(x)dx =

� π

0

� π/2

−π/2

cos(mθ) cos(nθ)dθ (5.154)

sin(mθ) sin(nθ)dθ (5.155)

which is precisely the normalization integral we perform for the particle in a box state assuming

the width **of** the box was π. For more generic applications, we can scale θ and its range to any

range.

1 There are at least 10 ways to spell Tchebychev’s last name Tchebychev, Tchebyshev, Chebyshev are the

most common, as well as Tchebysheff, Tchebycheff, Chebysheff, Chevychef, . . .

141

The way we use this is to use the φn = N sin nx basis functions as a finite basis and truncate

any expansion in this basis at some point. For example, since we are usually interested in low

lying energy states, setting an energy cut-**of**f to basis is exactly equivalent to keeping only the

lowest ncut states. The kinetic energy part **of** the Hamiltonian is diagonal in this basis, so we get

that part for free. However, the potential energy part is not diagonal in the φn = N sin nx basis,

so we have to compute its matrix elements:

�

Vnm = φn(x)V (x)φm(x)dx (5.156)

To calculate this integral, let us first realize that [V, x] = 0, so the eigenstates **of** x are also

eigenstates **of** the potential. Taking matrix elements in the finite basis,

xnm = N 2

�

φn(x)xφm(x)dx,

and diagonalizing it yields a finite set **of** ”position” eigenvalues, {xi} and a transformation for

converting between the ”position representation” and the ”basis representation”,

Tin = 〈xi|φn〉,

which is simply a matrix **of** the basis functions evaluated at each **of** the eigenvalues. The special

set **of** points defined by the eigenvalues **of** the position operator are the Gaussian quadrature

points over some finite range.

This proceedure termed the ”discrete variable representation” was developed by Light and

coworkers in the 80’s and is a very powerful way to generate coordinate representations **of** Hamiltonian

matrixes. Any matrix in the basis representation (termed the FBR for finite basis representation)

can be transformed to the discrete variable representation (DVR) via the transformation

matrix T . Moreover, there is a 1-1 correspondency between the number **of** DVR points and the

number **of** FBR basis functions. Here we have used only the Tchebychev functions. One can

generate DVRs for any set **of** orthogonal polynomial function. The Mathematica code below generates

the required transformations, the points, the eigenvalues **of** the second-derivative operator,

and a set **of** quadrature weights for the Tchebychev sine functions over a specified range:

dv2fb[DVR_, T_] := T.DVR.Transpose[T];

fb2dv[FBR_, T_] := Transpose[T].FBR.T;

tcheby[npts_, xmin_, xmax_] := Module[{pts, fb, del},

del = xmax - xmin;

pts = Table[i*del*(1/(npts + 1)) + xmin, {i, npts}] // N;

fbrke = Table[(i*(Pi/del))^2, {i, npts}] // N;

w = Table[del/(npts + 1), {i, npts}] // N;

T = Table[

Sqrt[2.0/(npts + 1)]*Sin[(i*j)*Pi/(npts + 1)],

{i, npts}, {j, npts}] // N;

Return[{pts, T, fbrke, w}]

]

To use this, we first define a potential surface, set up the Hamiltonian matrix, and simply

diagonalize. For this example, we will take the same double well system described above and

compare results and timings.

142

V[x_] := a*(x^4 - x^2);

cmm = 8064*27.3;

params = {a -> 0.1, m -> 1836};

{x, T, K, w} = tcheby[100, -1.3, 1.3];

Kdvr = (fb2dv[DiagonalMatrix[K], T]*m)/2 /. params;

Vdvr = DiagonalMatrix[V[x]] /. params;

Hdvr = Kdvr + Vdvr;

tt = Timing[{w, psi} = Transpose[

Sort[Transpose[Eigensystem[Hdvr]]]]];

Print[tt]

(Select[w*cmm , (# < 3000) &]) // TableForm

This code sets up the DVR points x, the transformation T and the FBR eigenvalues K using

the tcheby[n,xmin,xmax] Mathematica module defined above. We then generate the kinetic

energy matrix in the DVR using the transformation

and form the DVR Hamiltonian

KDV R = T † KF BRT

HDV R = KDV R + VDV R.

The eigenvalues and eigenvectors are computed via the Eigensystem[] routine. These are then

sorted according to their energy. Finally we print out only those states with energy less than

3000 cm −1 and check how long it took. On my 300 MHz G3 laptop, this took 0.3333 seconds to

complete. The first few **of** these are shown in Table 5.2 below. For comparison, each Numerov

iteration took roughly 1 second for each trial function. Even then, the eigenvalues we found are

probabily not as accurate as those computed here.

Table 5.2: Eigenvalues for double well potential computed via DVR and Numerov approaches

i ωi (cm −1 ) Numerov

1 -3946.574 -3946.59

2 -3943.7354 -3943.75

3 -1247.0974

4 -1093.5204

5 591.366

6 1617.424

143

5.5 Problems and Exercises

Exercise 5.2 Consider a harmonic oscillator **of** mass m and angular frequency ω. At time

t = 0, the state **of** this system is given by

|ψ(0)〉 = �

cn|φn〉 (5.157)

where the states |φn〉 are stationary states with energy En = (n + 1/2)¯hω.

n

1. What is the probability, P , that at a measurement **of** the energy **of** the oscillator at some

later time will yield a result greater than 2¯hω. When P = 0, what are the non-zero coefficients,

cn?

2. From now on, let only co and c1 be non zero. Write the normalization condition for |ψ(0)〉

and the mean value 〈H〉 **of** the energy in terms **of** co and c1. With the additional requirement

that 〈H〉 = ¯hω, calculate |co| 2 and |c1| 2 .

3. As the normalized state vector |ψ〉 is defined only to within an arbitrary global phase factor,

as can fix this factor by setting co to be real and positive. We set c1 = |c1|e iφ . We assume

also that 〈H〉 = ¯hω and show that

Calculate φ.

〈x〉 = 1

�

2

¯h

.

mω

(5.158)

4. With |ψ〉 so determined, write |ψ(t)〉 for t > 0 and calculate the value **of** φ at time t.

Deduce the mean **of** 〈x〉(t) **of** the position at time t.

Exercise 5.3 Find 〈x〉, 〈p〉, 〈x 2 〉 and 〈p 2 〉 for the ground state **of** a simple harmonic oscillator.

What is the uncertainty relation for the ground state.

Exercise 5.4 In this problem we consider the the interaction between molecule adsorbed on a

surface and the surface phonons. Represent the vibrational motion **of** the molecule (with reduced

mass µ) as harmonic with force constant K

and the coupling to the phonons as

Ho = −¯h2

2µ

∂2 K

+

∂x2 2 x2

(5.159)

H ′ = −x �

Vk cos(Ωkt) (5.160)

k

where Vk is the coupling between the molecule and phonon **of** wavevector k and frequency Ωk.

144

1. Express the total Hamiltonian as a displaced harmonic well. What happens to the well as

a function **of** time?

2. What is the Golden-Rule transition rate between the ground state and the nth excited state

**of** the system due to phonon interactions? Are there any restrictions as to which final state

can be reached? Which phonons are responsible for this process?

3. From now on, let the perturbing force be constant in time

H ′ = x �

k

Vk

(5.161)

where Vk is the interaction with a phonon with wavevector k. Use the lowest order level

**of** perturbation theory necessary to construct the transition probability between the ground

state and the second-excited state.

Exercise 5.5 Let

Show that the Harmonic Oscillator Hamiltonian is

X = ( mω

2¯h )1/2 x (5.162)

1

P = (

2¯hmω )1/2p (5.163)

H = ¯hω(P 2 + X 2 ) (5.164)

Now, define the operator: a † = X − iP . Show that a † acting on the harmonic oscillator ground

state is also an eigenstate **of** H. What is the energy **of** this state? Use a † to define a generating

relationship for all the eigenstates **of** H.

Exercise 5.6 Show that if one expands an arbitrary potential, V (x) about its minimum at xmin,

and neglects terms **of** order x 3 and above, one always obtains a harmonic well. Show that a

harmonic oscillator subject to a linear perturbation can be expressed as an unperturbed harmonic

oscillator shifted from the origin.

Exercise 5.7 Consider the one-dimensional Schrödinger equation with potential

�

m

V (x) = 2 ω2x2 x > 0

+∞ x ≤ 0

Find the energy eigenvalues and wavefunctions.

(5.165)

Exercise 5.8 An electron is contained inside a hard sphere **of** radius R. The radial components

**of** the lowest S and P state wavefunctions are approximately

ψS(x) ≈ sin(kr)

kr

ψP (x) ≈ cos(kr)

kr

− sin(kr)

(kr)

145

(5.166)

∂ψS(kr)

= . (5.167)

2 ∂(kr)

1. What boundary conditions must each state obey?

2. Using E = k 2 ¯h 2 /(2m) and the above boundary conditions, what are the energies **of** each

state?

3. What is the pressure exerted on the surface **of** the sphere if the electron is in the a.) S state,

b.) the P state. (Hint, recall from thermodynamics: dW = P dV = −(dE(R)/dR)dR.)

4. For a solvated e − in water, the S to P energy gap is about 1.7 eV. Estimate the size **of**

the the hard-sphere radius for the aqueous electron. If the ground state is fully solvated,

the pressure **of** the solvent on the electron must equal the pressure **of** the electron on the

solvent. What happens to the system when the electron is excited to the P -state from the

equilibrated S state? What happens to the energy gap between the S and P as a result **of**

this?

Exercise 5.9 A particle moves in a three dimensional potential well **of** the form:

V (x) =

�

∞ z 2 > a 2

mω 2

2 (x2 + y 2 ), otherwise

Obtain an equation for the eigenvalues and the associated eigenfunctions.

(5.168)

Exercise 5.10 A particle moving in one-dimension has a ground state wavefunction (not-normalized)

**of** the form:

ψo(x) = e −α4 x 4 /4

(5.169)

where α is a real constant with eigenvalue Eo = ¯h 2 α 2 /m. Determine the potential in which the

particle moves. (You do not have to determine the normalization.)

Exercise 5.11 A two dimensional oscillator has the Hamiltonian

H = 1

2 (p2 x + p 2 y) + 1

2 (1 + δxy)(x2 + y 2 ) (5.170)

where ¯h = 1 and δ

Figure 5.10: Ammonia Inversion and Tunneling

1. Using the Spartan electronic structure package (or any other one you have access to), build

a model **of** NH3, and determine its ground state geometry using various levels **of** ab initio

theory. Make a table **of** N − H bond lengths and θ = � H − N − H bond angles for the

equilibrium geometries as a function **of** at least 2 or 3 different basis sets. Looking in the

literature, find experimental values for the equilibrium configuration. Which method comes

closest to the experimental values? Which method has the lowest energy for its equilibrium

configuration.

2. Using the method which you deamed best in part 1, repeat the calculations you performed

above by systematically constraining the H − N − H bond angle to sample configurations

around the equilibrium configuration and up to the planar D3h configuration. Note, it may

be best to constrain two H − N − H angles and then optimize the bond lengths. Sample

enough points on either side **of** the minimum to get a descent potential curve. This is your

Born-Oppenheimer potential as a function **of** θ.

3. Defining the orgin **of** a coordinate system to be the θ = 120 o D3h point on the surface, fit

your ab initio data to the ”W”-potential

V (x) = αx 2 + βx 4

What are the theoretical values **of** α and β?

4. We will now use perturbation theory to compute the tunneling dynamics.

(a) Show that the points **of** minimum potential energy are at

� �1/2 α

xmin = ±

2β

(5.171)

and that the energy difference between the top **of** the barrier and the minimum energy

is given by

V = V (0) − V (xmin) (5.172)

= α2

4β

147

(5.173)

(b) We first will consider the barrier to be infinitely high so that we can expand the potential

function around each xmin. Show that by truncating the Taylor series expansion

above the (x − xmin) 2 terms that the potential for the left and right hand sides are

given by

VL = 2α (x + xmin) 2 − V

and

VR = 2α (x − xmin) 2 − V.

What are the vibrational energy levels for each well?

(c) The wavefunctions for the lowest energy states in each well are given by

with

ψ(x) = γ1/2

exp[−γ2

π1/4 2 (x ± xmin) 2 ]

γ =

� (4µα) 1/2

¯h

� 1/2

.

The energy levels for both sides are degenerate in the limit that the barrier height is

infinite. The total ground state wavefunction for this case is

Ψ(x) =

�

ψL(x)

ψR(x)

However, as the barrier height decreases, the degenerate states begin to mix causing

the energy levels to split. Define the ”high barrier” hamiltonian as

for x < 0 and

�

H = − ¯h2 ∂

2µ

2

+ VL(x)

∂x

H = − ¯h2 ∂

2µ

2

+ VR(x)

∂x

for x > 0. Calculate the matrix elements **of** H which mix the two degenerate left and

right hand ground state wavefunctions: i.e.

where

〈Ψ|H|Ψ〉 =

�

.

HRR HLR

HRL HLL

HRR = 〈ψR|H|ψR〉

, with similar definitions for HRL, HLL and HLR. Obtain numerical values **of** each

matrix element using the values **of** α and β you determined above (in cm −1 ). Use the

mass **of** a H atom for the reduced mass µ.

148

�

(d) Since the ψL and ψR basis functions are non-orthogonal, you will need to consider the

overlap matrix, S, when computing the eigenvalues **of** H. The eigenvalues for this

system can be determined by solving the secular equation

�

�

�

�

�

α − λ β − λS

β − λS α − λ

�

�

�

� = 0 (5.174)

�

where α = HRR = HLL and β = HLR = HRL (not to be confused with the potential

parameters above). Using Eq. 5.174, solve for λ and determine the energy splitting

in the ground state as a function the unperturbed harmonic frequency and the barrier

height, V . Calculate this splitting using the parameters you computed above. What is

the tunneling frequency? The experimental results is ∆E = 0.794cm −12 .

Exercise 5.14 Consider a system in which the Lagrangian is given by

L(qi, ˙qi) = T (qi, ˙qi) − V (qi) (5.175)

where we assume T is quadratic in the velocities. The potential is independent **of** the velocity

and neither T nor V carry any explicit time dependency. Show that

⎛

d

⎝

dt

�

⎞

∂L

˙qj − L⎠

= 0

∂ ˙qj

j

The constant quantity in the (. . .) defines a Hamiltonian, H. Show that under the assumed

conditions, H = T + V

Exercise 5.15 The Fermat principle in optics states that a light ray will follow the path, y(x)

which minimizes its optical length, S, through a media

S =

� x2,y2

x1,y1

n(y, x)ds

where n is the index **of** refraction. For y2 = y1 = 1 and −x1 = x2 = 1 find the ray-path for

1. n = exp(y)

2. n = a(y − yo) for y > yo

Make plots **of** each **of** these trajectories.

Exercise 5.16 In a quantum mechanical system there are gi distinct quantum states between

energy Ei and Ei + dEi. In this problem we will use the variational principle and Lagrange

multipliers to determine how ni particles are distributed amongst these states subject to the constraints

1. The number **of** particles is fixed:

n = �

2 From Molecular Structure and Dynamics, by W. Flygare, (Prentice Hall, 1978)

149

i

ni

2. the total energy is fixed �

niEi = E

We consider two cases:

i

1. For identical particles obeying the Pauli exclusion principle, the probability **of** a given configuration

is

WF D = �

i

gi

ni!(gi − ni)!

Show that maximizing WF D subject to the constraints above leads to

ni =

gi

e λ1+λ2Ei + 1

(5.176)

with the Lagrange multipliers λ1 = −Eo/kT and λ2 = 1/kT . Hint: try working with the

log W and use Sterling’s approximation in the limit **of** a large number **of** particles .

2. In this case we still consider identical particles, but relax the restriction on the fixed number

**of** particles in a given state. The probability for a given distribution is then

WBE = �

i

(ni + gi − 1)!

.

ni!(gi − 1)!

Show that by minimizing WBE subject to the constraints above leads to the occupation

numbers:

gi

ni =

eλ1+λ2Ei − 1

where again, the Lagrange multipliers are λ1 = −Eo/kT and λ2 = 1/kT . This yields the

Bose-Einstein statistics. Note: assume that gi ≫ 1

3. Photons satisfy the Bose-Einstein distribution and the constraint that the total energy is

constant. However, there is no constrain regarding the total number **of** photons. Show that

by eliminating the fixed number constraint leads to the foregoing result with λ1 = 0.

150

Bibliography

[1] C. A. Parr and D. G. Trular, J. Phys. Chem. 75, 1844 (1971).

[2] H. F. Schaefer III, J. Phys. Chem. 89, 5336 (1985).

[3] P. A. Whitlock and J. T. Muckermann, J. Chem. Phys. 61, 4624 (1974).

151

Chapter 6

**Quantum** **Mechanics** in 3D

In the next few lectures, we will focus upon one particular symmetry, the isotropy **of** free space.

As a collection **of** particles rotates about an arbitrary axis, the Hamiltonian does not change. If

the Hamoltonian does in fact depend explicitly upon the choice **of** axis, the system is “gauged”,

meaning all measurements will depend upon how we set up the coordinate frame. A Hamiltonian

with a potential function which depends only upon the coordinates, e.g. V = f(x, y, z), is gauge

invarient, meaning any measurement that I make will not depend upon my choice **of** reference

frame. On the other hand, if our Hamiltonian contains terms which couple one reference frame

to another (as in the case **of** non-rigid body rotations), we have to be careful in how we select

the “gauge”. While this sounds like a fairly specialized case, it turns out that many ordinary

phenimina depend upon this, eg. figure skaters, falling cats, floppy molecules. We focus upon

rigid body rotations first.

For further insight and information into the quantum mechanics **of** angular momentum, I

recommend the following texts and references:

1. Theory **of** Atomic Structure, E. Condon and G. Shortley. This is the classical book on

atomic physics and theory **of** atomic spectroscopy and has inspired generations since it

came out in 1935.

2. Angular Momentum–understanding spatial aspects in chemistry and physics, R. N. Zare.

This book is the text for the second-semester quantum mechanics at Stanford taught by

Zare (when he’s not out looking for Martians). It’s a great book with loads **of** examples in

spectroscopy.

3. **Quantum** Theory **of** Angular Momentum, D. A. Varshalovich, A. Moskalev, and V. Khersonskii.

Not to much physics in this book, but if you need to know some relation between

Wigner-D functions and Racah coefficients, or how to derive 12j symbols, this book is for

you.

First, we need to look at what happens to a Hamiltonian under rotation. In order to show that

H is invariant to any rotations, we need only to show that it is invarient under an infinitesimal

rotation.

152

6.1 **Quantum** Theory **of** Rotations

Let δ � φ by a vector **of** a small rotation equal in magnitude to the angle δφ directed long an

arbitrary axis. Rotating the system by δ � φ changes the direction vectors �rα by δ�rα.

δ�rα = δ � φ × �rα

Note that the × denotes the vector “cross” product. Since we will be using cross-products

through out these lectures, we pause to review the operation.

A cross product between two vectors is computed as

�c = �a × �b �

�

� î ˆj kˆ

�

= �

�

�

ai aj ak

bi bj bk

�

�

�

�

�

�

�

= î(ajbk − bjak) − ˆj(aibk − biak) + ˆ k(aibj − biaj)

(6.1)

= ɛijkajbk (6.2)

Where ɛijk is the Levi-Cevita symbol or the “anti-symmetric unit tensor” defined as

⎧

⎪⎨ 0 if any **of** the indices are the same

ɛijk = 1 for even permuations **of** the indices

⎪⎩

−1 for odd permutations **of** the indices

(Note that we also have assumed a “summation convention” where by we sum over all repeated

indices. Some elementary properties are ɛiklɛikm = δlm and ɛiklɛikl = 6.)

So, an arbitrary function ψ(r1, r2, · · ·) is transformed by the rotation into:

ψ1(r1 + δr1, r2 + δr2, · · ·) = ψ(r1, r2, · · ·) + �

δra · � ∇ψa

Thus, we conclude, that the operator

= ψ(r1, r2, · · ·) + �

δ� φ × ra · � ∇aψa

=

1 + δ� φ · �

�ra × � ∇a

a

�

1 + δ� φ · �

�ra × � �

∇a

is the operator for an infintesimal rotation **of** a system **of** particles. Since δφ is a constant, we

can show that this operator commutes with the Hamiltonian

�

�

[ �ra × � �

∇a , H] = 0 (6.6)

a

This implies then a particular conservation law related to the isotropy **of** space. This is **of** course

angular momentum so that

�

�

�ra × � �

∇a

(6.7)

a

153

a

a

a

ψa

(6.3)

(6.4)

(6.5)

must be at least proportional to the angular momentum operator, L. The exact relation is

which is much like its classical counterpart

¯hL = �r × �p = −i¯h�r × � ∇ (6.8)

L = 1

�r × �v. (6.9)

m

The operator is **of** course a vector quantity, meaning that is has direction. The components **of**

the angular momentum vector are:

¯hLx = ypz − zpy

¯hLy = zpx − zpz

¯hLz = xpy − ypx

¯hLi = ɛijkxjpk

(6.10)

(6.11)

(6.12)

(6.13)

For a system in a external field, antgular momentum is in general not conserved. However, if

the field posesses spherical symmetry about a central point, all directions in space are equivalent

and angular momentum about this point is conserved. Likewise, in an axially symmetric field,

motion about the axis is conserved. In fact all the conservation laws which apply in classical

mechanics have quantum mechanical analogues.

We now move on to compute the commutation rules between the Li operators and the x and

p operators First we note:

In short hand:

[Lx, x] = [Ly, y] = [Lz, z] = 0 (6.14)

[Lx, y] = 1

¯h ((ypz − zpy)y − y(ypz − zpy)) = − z

¯h [py, y] = iz (6.15)

[Li, xk] = iɛiklxl

We need also to know how the various components commute with one another:

¯h[Lx, Ly] = Lx(zpx − xpz) − (zpx − xpz)Lx

(6.16)

(6.17)

= (Lxz − zLx)px − x(Lxpz − pzLx) (6.18)

= −iypx + ixpy

154

(6.19)

Which we can summarize as

= i¯hLz

[Ly, Lz] = iLx

[Lz, Lx] = iLy

[Lx, Ly] = iLz

[Li, Lj] = iɛijkLk

Now, denote the square **of** the modulus **of** the total angular momentum by L 2 , where

L 2 = L 2 x + L 2 y + L 2 z

Notice that this operator commutes with all the other Lj operators,

For example:

also,

Thus,

(6.20)

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

[L 2 , Lx] = [L 2 , Ly] = [L 2 , Lz] = 0 (6.26)

[L 2 x, Lz] = Lx[Lx, Lz] + [Lx, Lz]Lx = −i(LxLy + LyLx) (6.27)

[L 2 y, Lz] = i(LxLy + LyLx) (6.28)

[L 2 , Lz] = 0 (6.29)

Thus, I can measure L 2 and Lz simultaneously. (Actually I can measure L 2 and any one component

Lk simultaneously. However, we ueually pick this one as the z axis to make the math

easier, as we shall soon see.)

A consequence **of** the fact that Lx, Ly, and Lz do not commute is that the angular momentum

vector � L can never lie exactly along the z axis (or exactly along any other

�

axis for that matter).

We can interpret this in a classical context as a vector **of** length |L| = ¯h L(L + 1) with the Lz

component being ¯hm. The vector is then constrained to lie in a cone as shown in Fig. ??. We

will take up this model at the end **of** this chapter in the semi-classical context.

It is also convienent to write Lx and Ly as a linear combination

L+ = Lx + iLyL− = Lx − iLy

(Recall what we did for Harmonic oscillators?) It’s easy to see that

[L+, L−] = 2Lz

155

(6.30)

(6.31)

Figure 6.1: Vector model for the quantum angular momentum state |jm〉, which is represented

here by the vector j which precesses about the z axis (axis **of** quantzation) with projection m.

Z

Y

Likewise:

m

θ

|j|=(j (j + 1))

1/2

X

[Lz, L+] = L+

[Lz, L−] = −L−

L 2 = L+L− + L 2 z − Lz = L−L+ + L 2 z + Lz

(6.32)

(6.33)

(6.34)

We now give some frequently used expressions for the angular momentum operator for a

single particle in spherical polar coordinates (SP). In SP coordinates,

It’s easy and straightforward to demonstrate that

and

L± = e ±φ

x = r sin θ cos φ (6.35)

y = r sin θ sin φ (6.36)

z = r cos θ (6.37)

Lz = −i ∂

∂φ

�

± ∂

�

∂

+ i cot θ

∂θ ∂φ

156

(6.38)

(6.39)

Thus,

L 2 = − 1

�

1 ∂

sin θ sin θ

2

�

∂ ∂

+ sin θ

∂φ2 ∂θ ∂θ

which is the angular part **of** the Laplacian in SP coordinates.

∇ 2 1

=

r2 �

sin θ

sin θ

∂ ∂ ∂ ∂ 1 ∂

r2 + sin θ +

∂r ∂r ∂θ ∂θ sin θ

2

∂φ2 �

= 1

r 2

∂ ∂

r2

∂r ∂r

1

− L2

r2 In other words, the kinetic energy operator in SP coordinates is

− ¯h2

2m ∇2 = − ¯h2

�

1

2m r2 �

∂ ∂ 1

r2 − L2

∂r ∂r r2 6.2 Eigenvalues **of** the Angular Momentum Operator

Using the SP form

(6.40)

(6.41)

(6.42)

(6.43)

Lzψ = i ∂ψ

∂φ = lzψ (6.44)

Thus, we conclude that ψ = f(r, θ)e ilzφ . This must be single valued and thus periodic in φ with

period 2π. Thus,

Thus, we write the azimuthal solutions as

which are orthonormal functions:

� 2π

0

lz = m = 0, ±1, ±2, · · · (6.45)

Φm(φ) = 1

√ 2π e imφ

(6.46)

Φ ∗ m(φ)Φm ′(φ)dφ = δmm ′ (6.47)

In a centrally symmetric case, stationary states which differ only in their m quantum number

must have the same energy.

We now look for the eigenvalues and eigenfunctions **of** the L 2 operator belonging to a set **of**

degenerate energy levels distinguished only by m. Since the +z−axis is physically equivalent to

the −z−axis, for every +m there must be a −m. Let L denote the greatest possible m for a

given L 2 eigenstate. This upper limit must exist because **of** the fact that L 2 − L 2 z = L 2 x + L 2 y is a

operator for an essentially positive quantity. Thus, its eigenvalues cannot be negative. We now

apply LzL± to ψm.

Lz(L±ψm) = (Lz ± 1)(L±ψm) = (m ± 1)(L±ψm) (6.48)

157

(note: we used [Lz, L±] = ±L± ) Thus, L±ψm is an engenfunction **of** Lz with eigenvalue m ± 1.

i.e.

ψm+1 ∝ L+ψm

ψm−1 ∝ L−ψm

If m = l then, we must have L+ψl = 0. Thus,

(6.49)

(6.50)

L−L+ψl = (L 2 − L 2 z − Lz)ψl = 0 (6.51)

L 2 ψl = (L 2 z + Lz)ψl = l(l + 1)ψl

(6.52)

Thus, the eigenvalues **of** L 2 operator are l(l + 1) for l any positive integer (including 0). For a

given value **of** l, the component Lz can take values

l, l − 1, · · · , 0, −l (6.53)

or 2l + 1 different values. Thus an energy level with angular momentum l has 2l + 1 degenerate

states.

6.3 Eigenstates **of** L 2

Since l ansd m are the good quantum numbers, we’ll denote the eigenstates **of** L 2 as

This we will **of**ten write in short hand after specifying l as

Since L 2 = L+L− + L 2 z − Lz, we have

Also, note that

thus we have

〈m|L 2 |m〉 = m 2 − m − �

L 2 |lm〉 = l(l + 1)|lm〉. (6.54)

L 2 |m〉 = l(l + 1)|m〉. (6.55)

m ′

〈m|L+|m ′ 〉〈m ′ |L−|m〉 = l(l + 1) (6.56)

〈m − 1|L−|m〉 = 〈m|L+|m − 1〉 ∗ , (6.57)

|〈m|L+|m − 1〉| 2 = l(l + 1) − m(m − 1) (6.58)

Choosing the phase (Condon and Shortly phase convention) so that

〈m|L+|m − 1〉 =

〈m − 1|L−|m〉 = 〈m|L+|m − 1〉 (6.59)

�

�

l(l + 1) − m(m − 1) = (l + m)(l − m + 1) (6.60)

Using this relation, we note that

〈m|Lx|m − 1〉 = 〈m − 1|Lx|m〉 = 1�

(l + m)(l − m + 1) (6.61)

2

〈m|Ly|m − 1〉 = 〈m − 1|Lx|m〉 = −i�

(l + m)(l − m + 1) (6.62)

2

Thus, the diagonal elements **of** Lx and Ly are zero in states with definite values **of** 〈Lz〉 = m.

158

6.4 Eigenfunctions **of** L 2

The wavefunction **of** a particle is not entirely determined when l and m are presribed. We still

need to specify the radial component. Thus, all the angular momentum operators (in SP coords)

contain and explicit r dependency. For the time, we’ll take r to be fixed and denote the angular

momentum eigenfunctions in SP coordinates as Ylm(θ, φ) with normalization

�

|Ylm(θ, φ)| 2 dΩ (6.63)

where dΩ = sin θdθdφ = d(cos θ)dφ and the integral is over all solid angles. Since we can

determine common eigenfunctions for L 2 and Lz, there must be a separation **of** variables, θ and

φ, so we seek solutions **of** the form:

The normalization requirement is that

and I require

I thus seek solution **of**

� 1

sin θ

i.e.

Ylm(θ, φ) = Φm(φ)Θlm(θ) (6.64)

� π

0

� 2π � π

0

0

∂ ∂ 1

sin θ +

∂θ ∂θ sin2 θ

|Θlm(θ)| 2 sin θdθ = 1 (6.65)

Y ∗

l ′ m ′YlmdΩ = δll ′δmm ′. (6.66)

∂2 ∂φ2 �

ψ + l(l + 1)ψ = 0 (6.67)

�

1 ∂ ∂ m2

sin θ −

sin θ ∂θ ∂θ sin2 �

+ l(l + 1) Θlm(θ) = 0 (6.68)

θ

which is well known from the theory **of** spherical harmonics.

Θlm(θ) = (−1) m i l

�

�

�

� (2l + 1)(l − m)!

P

2(l − m)!

m

l (cos θ) (6.69)

for m > 0. Where P m

l are associated Legendre Polynomials. For m < 0 we get

Θl,−|m| = (−1) m Θl,|m|

(6.70)

Thus, the angular momentum eigenfunctions are the spherical harmonics, normalized so that

the matrix relations defined above hold true. The complete expression is

Ylm = (−1) (m+|m|)/2 i l

� 2l + 1

4π

�1/2 (l − |m|)!

P

(l + |m|)!

|m|

l (cos θ)e imφ

159

(6.71)

Table 6.1: Spherical Harmonics (Condon-Shortley Phase convention.

1

Y00 = √

4π

� �1/2 3

Y1,0 = cos(θ)

4π

� �1/2 3

Y1,±1 = ∓ sin(θ)e

8π

±iφ

�

5

Y2,±2 = 3

96π sin2 θe ∓iφ

�

5

Y2,±1 = ∓3 sin θ cos θeiφ

24π

�

�

5 3

Y2,0 =

4π 2 cos2 θ − 1

�

2

These can also be generated by the SphericalHarmonicY[l,m,θ,φ] function in Mathematica.

Figure 6.2: Spherical Harmonic Functions for up to l = 2. The color indicates the phase **of** the

function.

160

For the case **of** m = 0,

Yl0 = i l

� 2l + 1

4π

� 1/2

Pl(cos θ) (6.72)

Other useful relations are in cartesian form, obtained by using the relations

cos θ = z

, (6.73)

r

sin θ cos φ = x

, (6.74)

r

and

sin θ sin φ = y

. (6.75)

r

Y1,0 =

Y1,1 =

� �1/2 3 z

4π r

� �1/2 3 x + iy

8π r

� �1/2 3 x − iy

Y1,−1 =

8π r

The orthogonality integral **of** the Ylm functions is given by

� 2π � π

Another useful relation is that

0

0

(6.76)

(6.77)

(6.78)

Y ∗

lm(θ, φ)Yl ′ m ′(θ, φ) sin θdθdφ = δll ′δmm ′. (6.79)

Yl,−m = (−1) m Y ∗

lm. (6.80)

This relation is useful in deriving real-valued combinations **of** the spherical harmonic functions.

Exercise 6.1 Demonstrate the following:

1. [L+, L 2 ] = 0

2. [L−, L 2 ] = 0

Exercise 6.2 Derive the following relations

�

�

�

ψl,m(θ, φ) = � (l + m)!

(2l!)(l − m)! (L−) l−m ψl,l(θ, φ)

and

�

�

�

ψl,m(θ, φ) = � (l − m)!

(2l!)(l + m)! (L+) l+m ψl,−l(θ, φ)

where ψl,m = Yl,m are eigenstates **of** the L 2 operator.

161

6.5 Addition theorem and matrix elements

In the quantum mechanics **of** rotations, we will come across integrals **of** the general form

�

Y ∗

l1m1 Yl2m2Yl3m3dΩ

or �

Y ∗

l1m1 Pl2Yl3m3dΩ

in computing matrix elements between angular momentum states. For example, we may be

asked to compute the matrix elements for dipole induced transitions between rotational states **of**

a spherical molecule or between different orbital angular momentum states **of** an atom. In either

case, we need to evaluate an integral/matrix element **of** the form

�

〈l1m1|z|l2m2〉 =

�

Realizing that z = r cos θ = r 4π/3Y10(θ, φ), Eq. 6.87 becomes

〈l1m1|z|l2m2〉 =

�

4π

3 r

�

Y ∗

l1m1 zYl2m2dΩ (6.81)

Y ∗

l1m1 Y10Yl2m2dΩ (6.82)

Integrals **of** this form can be evaluated by group theoretical analysis and involves the introduction

**of** Clebsch-Gordan coefficients, CLM 1

l1m1l2m2 which are tabulated in various places or can be

computed using Mathematica. In short, some basic rules will always apply.

1. The integral will vanish unless the vector sum **of** the angular momenta sums to zero.

i.e.|l1 − l3| ≤ l2 ≤ (l1 + l3). This is the “triangle” rule and basically means you have to be

able make a triangle with length **of** each side being l1, l2, and l3.

2. The integral will vanish unless m2 + m3 = m1. This reflects the conservation **of** the z

component **of** the angular momentum.

3. The integral vanishes unless l1 + l2 + l3 is an even integer. This is a parity conservation

law.

So the general proceedure for performing any calculation involving spherical harmonics is to first

check if the matrix element violates any **of** the three symmetry rules, if so, then the answer is 0

and you’re done. 2

To actually perform the integration, we first write the product **of** two **of** the Ylm’s as a

Clebsch-Gordan expansion:

Yl1m1Yl2m2 = �

�

�

�

� (2l1 + 1)(2l2 + 1)

4π(2L + 1) CL0

LM

l10l20C LM

l1m1l2m2 YLM. (6.83)

1 Our notation is based upon Varshalovich’s book. There at least 13 different notations that I know **of** for

expressing these coefficients which I list in a table at the end **of** this chapter.

2 In Mathematica, the Clebsch-Gordan coefficients are computed using the function

ClebschGordan[j1,m1,j2,m2,j,m] for the decomposition **of** |jm〉 in to |j1, m1〉 and |j2, m2〉.

162

We can use this to write

�

Y ∗

lmYl1m2Yl2m2dΩ = �

�

�

�

� (2l1 + 1)(2l2 + 1)

4π(2L + 1) CL0

LM

�

�

�

= �

�

LM

(2l1 + 1)(2l2 + 1)

4π(2L + 1) CL0

�

�

�

= � (2l1 + 1)(2l2 + 1)

4π(2l + 1)

l10l20C LM

�

l1m1l2m2

l10l20C LM

l1m1l2m2 δlLδmM

C l0

l10l20C lm

l1m1l2m2

Y ∗

lmYLMdΩ

(6.84)

In fact, the expansion we have done above for the product **of** two spherical harmonics can be

inverted to yield the decomposition **of** one angular momentum state into a pair **of** coupled angular

momentum states, such as would be the case for combining the orbital angular momentum **of** a

particle with, say, its spin angular momentum. In Dirac notation, this becomes pretty apparent

|LM〉 = �

〈l1m1l2m2|LM〉|l1m1l2m2〉 (6.85)

m1m2

where the state |l1m1l2m2〉 is the product **of** two angular momentum states |l1m1〉 and |l2m2〉.

The expansion coefficients are the Clebsch-Gordan coefficients

C LM

l1m1l2m2 = 〈l1m1l2m2|LM〉 (6.86)

Now, let’s go back the problem **of** computing the dipole transition matrix element between

two angular momentum states in Eq. 6.87. The integral we wish to evaluate is

�

〈l1m1|z|l2m2〉 =

Y ∗

l1m1 zYl2m2dΩ (6.87)

and we noted that z was related to the Y10 spherical harmonic. So the integral over the angular

coordinates involves:

�

Y ∗

l1m1 Y10Yl2m2dΩ. (6.88)

First, we evaluate which matrix elements are going to be permitted by symmetry.

1. Clearly, by the triangle inequality, |l1 − l2| = 1. In other words, we change the angular

momentum quantum number by only ±1.

2. Also, by the second criteria, m1 = m2

3. Finally, by the third criteria: l1 + l2 + 1 must be even, which again implies that l1 and l2

differ by 1.

Thus the integral becomes

�

Y ∗

l+1,mY10Ylm =

�

�

�

� (2l + 1)(2 + 1)

4π(2l + 3) Cl+1,0 l010 C1m l+1,ml0

163

(6.89)

From tables,

So

Thus,

�

C l+1,0

l010

C 1m

l+1,ml0 = −

= −

� √ �

2 (1 + l)

√ √

2 + 2 l 3 + 2 l

�√ √ √ �

2 1 + l − m 1 + l + m

√ √

2 + 2 l 3 + 2 l

C l+1,0

l010 C1m

l+1,ml0 = 2 (1 + l) √ 1 + l − m √ 1 + l + m

(2 + 2 l) (3 + 2 l)

Y ∗

l+1,mY10YlmdΩ =

� �

�

3 �

� (l + m + 1)(l − m + 1)

4π (2l + 1)(2l + 3)

Finally, we can construct the matrix element for dipole-transitions as

(6.90)

�

�

�

〈l1m1|z|l2m2〉 = r�

(l + m + 1)(l − m + 1)

δl1±1,l2δm1,m2. (6.91)

(2l + 1)(2l + 3)

Physically, this make sense because a photon carries a single quanta **of** angular momentum. So

in order for molecule or atom to emit or absorb a photon, its angular momentum can only change

by ±1.

Exercise 6.3 Verify the following relations

�

�

Y ∗

l+1,m+1Y11Ylmdω =

Y ∗

l−1,m−1Y11Ylmdω = −

�

� �

�

3 �

� (l + m + 1)(l + m + 2)

8π 2l + 1)(2l + 3)

� �

�

3 �

� (l − m)(l − m − 1)

8π 2l − 1)(2l + 1)

Y ∗

lmY00YlmdΩ = 1

√ 4π

(6.92)

(6.93)

(6.94)

6.6 Legendre Polynomials and Associated Legendre Polynomials

Ordinary Legendre polynomials are generated by

Pl(cos θ) = 1

2ll! d l

(d cos θ) l (cos2 θ − 1) l

164

(6.95)

i.e. (x = cos θ)

and satisfy

Pl(x) = 1

2l ∂

l!

l

∂x l (x2 − 1) l

(6.96)

� �

1 ∂ ∂

sin θ + l(l + 1) Pl = 0 (6.97)

sin θ ∂θ ∂θ

The Associated Legendre Polynomials are derived from the Legendre Polynomials via

P m

l (cos θ) = sin m θ

∂m (∂ cos θ) m Pl(cos θ) (6.98)

6.7 **Quantum** rotations in a semi-classical context

Earlier we established the fact that the angular momentum vector can never exactly lie on a

single spatial axis. By convention we take the quantization axis to be the z axis, but this is

arbitrary and we can pick any axis as the quantization axis, it is just that picking the z axis

make the mathematics much simpler. Furthermore, we established that the maximum length

the angular momentum vector can have along the z axis is the eigenvalue **of** Lz �

when m = l,

so 〈lz〉 = l which is less than l(l + 1). Note, however, we can write the eigenvalue **of** L2 as

l 2 (1+1/l 2 ) . As l becomes very large the eigenvalue **of** Lz and the eigenvalue **of** L 2 become nearly

identical. The 1/l term is in a sense a quantum mechanical effect resulting from the uncertainty

in determining the precise direction **of** � L.

We can develop a more quantative model for this by examining both the uncertainty product

and the semi-classical limit **of** the angular momentum distribution function. First, recall, that if

we have an observable, A then the spread in the measurements **of** A is given by the varience.

∆A 2 = 〈(A − 〈A〉) 2 〉 = 〈A 2 〉 − 〈A〉 2 . (6.99)

In any representation in which A is diagonal, ∆A 2 = 0 and we can determine A to any level **of**

precision. But if we look at the sum **of** the variances **of** lx and ly we see

∆L 2 x + ∆L 2 y = l(l + 1) − m 2 . (6.100)

So for a fixed value **of** l and m, the sum **of** the two variences is constant and reaches its minimum

when |m| = l corresponding to the case when the vector points as close to the ±z axis as it

possible can. The conclusion we reach is that the angular momentum vector lies somewhere in a

cone in which the apex half-angle, θ satisfies the relation

m

cos θ = �

(6.101)

l(l + 1)

which we can varify geometrically. So as l becomes very large the denominator becomes for m = l

l

�

l(l + 1) =

1

� → 1 (6.102)

1 1 + 1/l2 165

and θ = 0,cooresponding to the case in which the angular momentum vector lies perfectly along

the z axis.

Exercise 6.4 Prove Eq. 6.100 by writing 〈L 2 〉 = 〈L 2 x〉 + 〈L 2 y〉 + 〈L 2 z〉.

To develop this further, let’s look at the asymptotic behavour **of** the Spherical Harmonics at

large values **of** angular momentum. The angular part **of** the Spherical Harmonic function satisfies

�

1 ∂ ∂

m2

sin θ + l(l + 1) −

sin θ ∂θ ∂θ sin2 �

Θlm = 0 (6.103)

θ

For m = 0 this reduces to the differential equation for the Legendre polynomials

� �

2 ∂ ∂

+ cot θ + l(l + 1) Pl(cos θ) = 0 (6.104)

∂θ2 ∂θ

If we make the substitution

Pl(cos θ) =

χlθ

(sin θ) 1/2

(6.105)

then we wind up with a similar equation for χl(θ)

�

2 ∂

∂θ2 + (l + 1/2)2 + csc2 �

θ

χl = 0. (6.106)

4

For very large l, the l + 1/2 term dominates and we can ignore the cscθ term everywhere except

for angles close to θ = 0 or θ = π. If we do so then our differential equation becomes

� �

2 ∂

+ (l + 1/2)2 χl = 0, (6.107)

∂θ2 which has the solution

χl(θ) = Al sin ((l + 1/2)θ + α) (6.108)

where Al and α are constants we need to determine from the boundary conditions **of** the problem.

For large l and for θ ≫ l −1 and π − θ ≫ l −1 one obtains

Similarly,

Ylo(θ, φ) ≈

sin((l + 1/2)θ + α)

Pl(cos θ) ≈ Al

(sin θ) 1/2 . (6.109)

� l + 1/2

2π

� 1/2

so that the angular probability distribution is

� �

l + 1/2

|Ylo| 2 =

2π

A 2 l

sin((l + 1/2)θ + α)

Al

(sin θ) 1/2 . (6.110)

sin2 ((l + 1/2)θ + α)

. (6.111)

sin θ

166

When l is very large the sin 2 ((l + 1/2)θ) factor is extremely oscillatory and we can replace it

by its average value **of** 1/2. Then, if we require the integral **of** our approximation for |Yl0| 2 to be

normalized, one obtains

|Yl0| 2 =

1

2π 2 sin(θ)

(6.112)

which holds for large values **of** l and all values **of** θ except for theta = 0 or θ = π.

We can also recover this result from a purely classical model. In classical mechanics, the

particle moves in a circular orbit in a plane perpendicular to the angular momentum vector. For

m = 0 this vector lies in the xy plane and we will define θ as the angle between the particle and

the z axis, φ as the azimuthal angle **of** the angular momentum vector in the xy plane. Since the

particles speed is uniform, its distribution in θ is uniform. Thus the probability **of** finding the

particle at any instant in time between θ and θ + dθ is dθ/π. Furthermore, we have not specified

the azimuthal angle, so we assume that the probability distribution is also uniform over φ and

the angular probability dθ/π must be smeared over some band on the uniform sphere defined by

the angles θ and θ + dθ. The area **of** this band is 2π sin θdθ. Thus, we can define the “classical

estimate” **of** the as a probability per unit area

P (θ) = dθ

π

1

2π sin θ =

1

2π 2 sin θ

which is in agreement with the estimate we made above.

For m �= 0 we have to work a bit harder since the angular momentum vector is tilted out **of**

the plane. For this we define two new angles γ which is the azimuthal rotation **of** the particle’s

position about the L vector and α with is constrained by the length **of** the angular momentum

vector and its projection onto the z axis.

m m

cos α = � ≈

l(l + 1 l

The analysis is identical as before with the addition **of** the fact that the probability in γ (taken

to be uniform) is spread over a zone 2π sin θdθ. Thus the probability **of** finding the particle with

some angle θ is

P (θ) = dγ 1

dθ 2π2 sin θ .

Since γ is the dihedral angle between the plane containing z and l and the plane containing

l and r (the particle’s position vector), we can relate γ to θ and α by

Thus,

cos θ = cos α cos π π

+ sin α sin cos γ = sin α cos γ

2 2

sin θdθ = sin α sin γdγ.

This allows us to generalize our probability distribution to any value **of** m

|Ylm(θ, φ)| 2 =

=

1

2π2 sin α sin γ

1

2π 2 (sin 2 α − cos 2 θ) 1/2

167

(6.113)

(6.114)

Figure 6.3: Classical and **Quantum** Probability Distribution Functions for Angular Momentum.

1

0.8

0.6

0.4

0.2

1

0.8

0.6

0.4

0.2

1

0.8

0.6

0.4

0.2

»Y4,0» 2

0.5 1 1.5 2 2.5 3

»Y4,2» 2

0.5 1 1.5 2 2.5 3

»Y4,4» 2

0.5 1 1.5 2 2.5 3

1.5

1.25

1

0.8

0.6

0.4

0.2

1

0.8

0.6

0.4

0.2

1

0.75

0.5

0.25

»Y10,0» 2

0.5 1 1.5 2 2.5 3

»Y10,2» 2

0.5 1 1.5 2 2.5 3

»Y10,10» 2

0.5 1 1.5 2 2.5 3

which holds so long as sin 2 α > cos 2 θ. This corresponds to the spatial region (π/2 − α) <

θ < π/2 + α. Outside this region, the distribution blows up and corresponds to the classically

forbidden region.

In Fig. 6.3 we compare the results **of** our semi-classical model with the exact results for l = 4

and l = 10. All in all we do pretty well with a semi-classical model, we do miss some **of** the

wiggles and the distribution is sharp close to the boundaries, but the generic features are all

there.

168

Table 6.2: Relation between various notations for Clebsch-Gordan Coefficients in the literature

Symbol Author

C jm

j1m1j2m2 Varshalovich a

S j1j2

jm1jm2 Wignerb Ajj1j2 mm1m2 Eckartc Cm1m j

2

Van der Weardend (j1j2m1m2|j1j2jm) Condon and Shortley e

C j1j2

jm (m1m2) Fock f

X(j, m, j1, j2, m1) Boys g

C(jm; m1m2) Blatt and Weisskopfh Cj1j2j m1m2m Beidenharni C(j1j2j, m1m2) Rosej �

�

j1 j2 j

m1 m2 m

Yutsis and Bandzaitisk 〈j1m1j2m2|(j1j2)jm〉 Fanol a.) D. A. Varschalovich, et al. **Quantum** Theory **of** Angular Momentum, (World Scientific, 1988).

b.) E. Wigner, Group theory, (Academic Press, 1959).

c.) C. Eckart “The application **of** group theory to the quantum dynamics **of** monatomic systems”,

Rev. Mod. Phys. 2, 305 (1930).

d.) B. L. Van der Waerden, Die gruppentheorische methode in der quantenmechanik, (Springer,

1932).

e.)E. Condon and G. Shortley, Theory **of** Atomic Spectra, (Cambridge, 1932).

f.) V. A. Fock, “ New Deduction **of** the Vector Model”, JETP 10,383 (1940).

g.)S. F. Boys, “Electronic wave functions IV”, Proc. Roy. Soc., London, A207, 181 (1951).

h.) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, (McGraw-Hill, 1952).

i.) L. C. Beidenharn, ”Tables **of** Racah Coefficients”, ONRL-1098 (1952).

j.) M. E. Rose, Multipole Fields, (Wiley 1955).

k.) A. P. Yusis and A. A. Bandzaitit, The Theory **of** Angular Momentum in Quanutm **Mechanics**,

(Mintus, Vilinus, 1965).

l.) U. Fano, “Statistical matrix techniques and their application to the directional correlation **of**

radiation,” US Nat’l Bureau **of** Standards, Report 1214 (1951).

169

6.8 Motion in a central potential: The Hydrogen Atom

(under development)

The solution **of** the Schrödinger equation for the hydrogen atom was perhaps the most significant

developments in quantum theory. Since it is one **of** the few problems in nature in which we

can derive an exact solution to the equations **of** motion, it deserves special attention and focus.

Perhaps more importantly, the hydrogen atomic orbitals form the basis **of** atomic physics and

quantum chemistry.

The potential energy function between the proton and the electron is the centrosymmetric

Coulombic potential

V (r) = − Ze2

r .

Since the potential is centrosymmetric and has no angular dependency the Hydrogen atom Hamiltonian

separates in to radial and angular components.

H = − ¯h2

2µ

� 1

r 2

∂ ∂ L2

r2 −

∂r ∂r ¯h 2 r2 �

+ e2

r

(6.115)

where L 2 is the angular momentum operator we all know and love by now and µ is the reduced

mass **of** the electron/proton system

µ = memp

me + mp

≈ me = 1

Since [H, L] = 0, angular momentum and one component **of** the angular momentum must be

constants **of** the motion. Since there are three separable degrees **of** freedom, we have one other

constant **of** motion which must correspond to the radial motion. As a consequence, the hydrogen

wavefunction is separable into radial and angular components

ψnlm = Rnl(r)Ylm(θ, φ). (6.116)

Using the Hamiltonian in Eq. 6.115 and this wavefunction, the radial Schrödinger equation reads

(in atomic units)

�

− ¯h2

�

1

2 r2 ∂ ∂ l(l + 1)

r2 −

∂r ∂r r2 �

− 1

�

Rnl(R) = ERnl(r) (6.117)

r

At this point, we introduce atomic units to make the notation more compact and drastically

simplify calculations. In atomic units, ¯h = 1 and e = 1. A list **of** conversions for energy, length,

etc. to SI units is listed in the appendix. The motivation is so that all **of** our numbers are **of**

order 1.

The kinetic energy term can be rearranged a bit

and the radial equation written as

�

− ¯h2

�

2 ∂ 2 ∂

+

2 ∂r2 r ∂r

1

r2 ∂ ∂

r2

∂r ∂r

= ∂2

∂r

2 ∂

+ 2 r ∂r

(6.118)

l(l + 1)

−

r2 �

− 1

�

Rnl(R) = ERnl(r) (6.119)

r

170

To solve this equation, we first have to figure out what approximate form the wavefunction must

have. For large values **of** r, the 1/r terms disappear and the asymptotic equation is

or

− ¯h2

2

∂ 2

∂r 2 Rnl(R) = ERnl(r) (6.120)

∂ 2 R

∂r 2 = α2 R (6.121)

where α = −2mE/¯h 2 . This differential equation we have seen before for the free particle, so the

solution must have the same form. Except in this case, the function is real. Furthermore, for

bound states with E < 0 the radial solution must go to zero as r → ∞, so **of** the two possible

asymptotic solutions, the exponentially damped term is the correct one.

R(r) ≡ e −αr

(6.122)

Now, we have to check if this is a solution everywhere. So, we take the asymptotic solution and

plug it into the complete equation:

Eliminating e −αr

α 2 e −αr + 2

r (−αe−αr ) + 2m

¯h 2

� �

2 e

+ E e

r −αr = 0. (6.123)

�

α 2 + 2mE

¯h 2

�

+ 1

� �

2 2me

2 − 2α = 0 (6.124)

r ¯h

For the solution to hold everywhere, it must also hold at r = 0, so two conditions must be met

which we defined above, and

α 2 = −2mE/¯h 2

� 2me 2

2 − 2α

¯h

�

(6.125)

= 0. (6.126)

If these conditions are met, then e −αr is a solution. This last equation also sets the length scale

**of** the system since

α = me 2 /¯h 2 = 1/ao

(6.127)

where ao is the Bohr radius. In atomic units, ao = 1. Likewise, the energy can be determined:

E = − ¯h2

2ma 2 o

= − ¯h2

me 2

e 2

2ao

In atomic units the ground states energy is E = −1/2hartree.

171

= − e2

. (6.128)

2ao

Finally, we have to normalize R

�

d 3 re −2αr � ∞

= 4π r

0

2 e −2αr dr (6.129)

The angular normalization can be absorbed into the spherical harmonic term in the total wavefunction

since Y00 = 1/ √ 4π. So, the ground state wavefunction is

ψn00 = Ne −r/ao Y00

(6.130)

The radial integral can be evaluated using Leibnitz’ theorem for differentiation **of** a definite

integral

Thus,

�

∂ b

� b

f(β, x)dx =

∂β a

a

� ∞

r

0

2 e −βr dr =

Exercise 6.5 Generalize this result to show that

� ∞

0

� ∞

0

= ∂2

∂β 2

= − ∂2

∂β 2

= 2

β 3

∂f(β, x)

dx (6.131)

∂β

∂ 2

∂β2 e−βrdr � ∞

0

1

β

r n e −βr dr = n!

β n+1

e −βr dr

Thus, using this result and putting it all together, the normalized radial wavefunction is

(6.132)

(6.133)

� �3/2 1

R10 = 2 e

ao

−r/ao . (6.134)

For the higher energy states, we examine what happens at r → 0. Using a similar analysis

as above, one can show that close in, the radial solution must behave like a polynomial

which leads to a general solution

R ≡ r l+1

R = r l+1 e −αr

∞�

asr

s=0

s .

172

The proceedure is to substitute this back into the Schrodinger equation and evaluate term by

term. In the end one finds that the energies **of** the bound states are (in atomic units)

En = − 1

2n 2

and the radial wavefunctions

� �l � �

2r 2

Rnl =

nao nao

� (n − l − 1)!

2n((n + l)!) 3

�

e −r/nao L 2l+1

� �

2r

n+1

nao

where the L b a are the associated Laguerre polynomials.

6.8.1 Radial Hydrogenic Functions

(6.135)

The radial wavefunctions for nuclei with atomic number Z are modified hydrogenic wavefunctions

with the Bohr radius scaled by Z. I.e a = ao/Z. The energy for 1 electron about a nucleus with

Z protons is

Some radial wavefunctions are

6.9 Spin 1/2 Systems

En = − Z2

n 2

1

2ao

= − Z2

n2 Ry

2

(6.136)

� �3/2

Z

R1s = 2 e

ao

−Zr/ao (6.137)

R2s = 1

� �3/2 �

Z

√ 1 −

2 ao

Zr

�

e

2ao

−Zr/2ao (6.138)

R2p = 1

2 √ � �5/2

Z

re

6 ao

−Zr/2ao (6.139)

In this section we are going to illustrate the various postulates and concepts we have been

developing over the past few weeks. Rather than choosing as examples problems which are

pedagogic (such as the particle in a box and its variations) or or chosen for theor mathematical

simplicity, we are going to focus upon systems which are physically important. We are going to

examine, with out much theoretical introduction, the case in which the state space is limited to

two states. The quantum mechanical behaviour **of** these systems can be varified experimentally

and, in fact, were and still are used to test various assumptions regarding quantum behaviour.

Recall from undergraduate chemistry that particles, such as the electron, proton, and so forth,

possess an intrinsic angular momentum, � S, called spin. This is a property which has no analogue

in classical mechanics. Without going in to all the details **of** angular momentum and how it gets

quantized (don’t worry, it’s a coming event!) we are going to look at a spin 1/2 system, such as

173

a neutral paramagnetic Ag atom in its ground electronic state. We are going to dispense with

treating the other variables, the nuclear position and momentum,the motion **of** the electrons,

etc... and focus only upon the spin states **of** the system.

The paramagnetic Ag atoms possess an electronic magnetic moment, � M. This magnetic

moment can couple to an externally applied magnetic field, � B, resulting on a net force being

applied to the atom. The potential energy in for this is

W = − � M. � B. (6.140)

We take this without further pro**of**. We also take without pro**of** that the magnetic moment and

the intrinsic angular momentum are proportional.

�M = γ � S (6.141)

The proportionality constant is the gyromagnetic ratio **of** the level under consideration. When

the atoms traverse through the magnetic field, they are deflected according to how their angular

momentum vector is oriented with the applied field.

Also, the total moment relative to the center **of** the atom is

�F = � ∇( � M. � B) (6.142)

� Γ = � M × � B. (6.143)

Thus, the time evolution **of** the angular momentum **of** the particle is

that it to say

∂

∂t � S = � Γ (6.144)

∂

∂t � S = γ � S × � B. (6.145)

Thus, the velocity **of** the angular momentum is perpendicular to � S and the angular momentum

vector acts like a gyroscope.

We can also show that for a homogeneous field the force acts parallel to z and is proportional

to Mz. Thus, the atoms are deflected according to how their angular momentum vector is oriented

with respect to the z axis. Experimentally, we get two distributions. Meaning that measurement

**of** Mz can give rise to two possible results.

6.9.1 Theoretical Description

We associate an observable, Sz, with the experimental observations. This has 2 eigenvalues, at

±¯h/2 We shall assume that the two are not degenerate. We also write the eigenvectors **of** Sz as

|±〉 corresponding to

Sz|+〉 = + ¯h

|+〉 (6.146)

2

174

with

and

The closure, or idempotent relation is thus

with

The most general state vector is

Sz|−〉 = + ¯h

|−〉 (6.147)

2

〈+|+〉 = 〈−|−〉 = 1 (6.148)

〈+|−〉 = 0. (6.149)

|+〉〈+| + |−〉〈−| = 1. (6.150)

|ψ〉 = α|+〉 + β|−〉 (6.151)

|α| 2 + |β| 2 = 1. (6.152)

In the |±〉 basis, the matrix representation **of** Sz is diagonal and is written as

Sz = ¯h

�

2

6.9.2 Other Spin Observables

1 0

0 −1

We can also measure Sx and Sy. In the |±〉 basis these are written as

and

Sx = ¯h

�

2

Sy = ¯h

�

2

0 1

1 0

0 i

−i 0

You can verify that the eigenvalues **of** each **of** these are ±¯h/2.

6.9.3 Evolution **of** a state

The Hamiltonian for a spin 1/2 particle in a B-field is given by

�

�

�

(6.153)

(6.154)

(6.155)

H = −γ|B|Sz. (6.156)

175

Where B is the magnitude **of** the field. This operator is time-independent, thus, we can solve the

Schrodinger Equation and see that the eigenvectors **of** H are also the eigenvectors **of** Sz. (This

the eigenvalues **of** Sz are “good quantum numbers”.) Let’s write ω = −γ|B| so that

H|+〉 = + ¯hω

|+〉 (6.157)

2

H|−〉 = − ¯hω

|−〉 (6.158)

2

Therefore there are two energy levels, E± = ±¯hω/2. The separation is proportional to the

magnetic field. They define a single “Bohr Frequency”.

6.9.4 Larmor Precession

Using the |±〉 states, we can write any arb. angular momentum state as

|ψ(0)〉 = cos( θ

2 )e−iφ/2 |+〉 + sin( θ

2 )e+iφ/2 |−〉 (6.159)

where θ and φ are polar coordinate angles specifing the directrion **of** the angular momentum

vector at a given time. The time evolution under H is

|ψ(0)〉 = cos( θ

2 )e−iφ/2 e −iE+t/¯h |+〉 + sin( θ

2 )e+iφ/2 e −iEmt/¯h |−〉, (6.160)

or, using the values **of** E+ and E−

|ψ(0)〉 = cos( θ

2 )e−i(φ+ωt)/2 |+〉 + sin( θ

2 )e+i(φ+ωt)/2 |−〉 (6.161)

In other words, I can write

θ(t) = θ (6.162)

φ(t) = φ + ωt. (6.163)

This corresponds to the precession **of** the angular momentum vector about the z axis at an angular

frequency **of** ω. More over, the expectation values **of** Sz, Sy, and Sx can also be computed:

〈Sz(t)〉 = ¯h/2 cos(θ) (6.164)

〈Sx(t)〉 = ¯h/2 sin(θ/2) cos(φ + ωt) (6.165)

〈Sy(t)〉 = ¯h/2 sin(θ/2) sin(φ + ωt) (6.166)

Finally, what are the “populations” **of** the |±〉 states as a function **of** time?

|〈+|ψ(t)〉| 2 = cos 2 (θ/2) (6.167)

|〈−|ψ(t)〉| 2 = sin 2 (θ/2) (6.168)

Thus, the populations do not change, neither does the normalization **of** the state.

176

6.10 Problems and Exercises

Exercise 6.6 A molecule (A) with orbital angular momentum S = 3/2 decomposes into two

products: product (B) with orbital angular momentum 1/2 and product (C) with orbital angular

momentum 0. We place ourselves in the rest frame **of** A) and angular momentum is conserved

throughout.

A 3/2 → B 1/2 + C 0

(6.169)

1. What values can be taken on by the relative orbital angular momentum **of** the two final

products? Show that there is only one possible value **of** the parity **of** the relative orbital

state is fixed. Would this result remain the same if the spin **of** A was 3/2?

2. Assume that A is initially in the spin state characterized by the eigenvalue ma¯h **of** its spin

component along the z-axis. We know that the final orbital state has a definite parity. Is it

possible to determine this parity by measuring the probabilities **of** finding B in either state

|+〉 or in state |−〉?

Exercise 6.7 The quadrupole moment **of** a charge distribution, ρ(r), is given by

Qij = 1

�

e

3(xixj − δijr 2 )ρ(r)d 3 r (6.170)

where the total charge e = � d3rρ(r). The quantum mechanical equivalent **of** this can be written

in terms **of** the angular momentum operators as

Qij = 1

�

r

e

2

�

3

2 (JiJj + JjJi) − δijJ 2

�

ρ(r)d 3 r (6.171)

The quadrupole moment **of** a stationary state |n, j〉, where n are other non-angular momentum

quantum numbers **of** the system, is given by the expectation value **of** Qzz in the state in which

m = j.

1. Evaluate

in terms **of** j and 〈r 2 〉 = 〈nj|r 2 |nj〉.

Qo = 〈Qzz〉 = 〈njm = j|Qzz|njm = j〉 (6.172)

2. Can a proton (j = 1/2) have a quadrupole moment? What a bout a deuteron (j = 1)?

3. Evaluate the matrix element

What transitions are induced by this operator?

〈njm|Qxy|nj ′ m ′ 〉 (6.173)

4. The quantum mechanical expression **of** the dipole moment is

po = 〈njm = j| r

e Jz|njm = j〉 (6.174)

Can an eigenstate **of** a Hamiltonian with a centrally symmetric potential have an electric

dipole moment?

177

Exercise 6.8 The σx matrix is given by

prove that

σx =

where α is a constant and I is the unit matrix.

�

0 1

1 0

�

, (6.175)

exp(iασx) = I cos(α) + iσx sin(α) (6.176)

Solution: To solve this you need to expand the exponential. To order α 4 this is

e iασx = I + iασx − α2

2 σ2 x − iα3

3! σ3 + α4

4! σ4 x + · · · (6.177)

Also, note that σx.σx = I, thus, σ2n x = I and σ2n+1 x = σx. Collect all the real terms and all the

imaginary terms:

e iασx =

�

I + I α2

2

+ I α4

4!

+ · · ·

These are the series expansions for cos and sin.

�

+ iσx

�

α − −i α3

3!

+ · · ·

�

(6.178)

e iασx = I cos(α) + iσx sin(α) (6.179)

Exercise 6.9 Because **of** the interaction between the proton and the electron in the ground state

**of** the Hydrogen atom, the atom has hyperfine structure. The energy matrix is **of** the form:

in the basis defined by

⎛

⎜

H = ⎜

⎝

A 0 0 0

0 −A 2A 0

0 2A −A 0

0 0 0 A

⎞

⎟

⎠

(6.180)

|1〉 = |e+, p+〉 (6.181)

|2〉 = |e+, p−〉 (6.182)

|3〉 = |e−, p+〉 (6.183)

|4〉 = |e−, p−〉 (6.184)

where the notation e+ means that the electron’s spin is along the +Z-axis, and e− has the spin

pointed along the −Z axis. i.e. |e+, p+〉 is the state in which both the electron spin and proton

spin is along the +Z axis.

1. Find the energy **of** the stationary states and sketch an energy level diagram relating the

energies and the coupling.

178

2. Express the stationary states as linear combinations **of** the basis states.

3. A magnetic field **of** strength B applied in the +Z direction and couples the |e+, p+〉 and

|e−, p−〉 states. Write the new Hamiltonian matrix in the |e±, p±〉 basis. What happens to

the energy levels **of** the stationary states as a result **of** the coupling? Add this information

to the energy level diagram you sketched in part 1.

Exercise 6.10 Consider a spin 1/2 particle with magnetic moment � M = γ � S. The spin space is

spanned by the basis **of** |+〉 and |−〉 vectors, which are eigenvectors **of** Sz with eigenvalues ±¯h/2.

At time t = 0, the state **of** the system is given by

|ψ(0)〉 = |+〉

1. If the observable Sx is measured at time t = 0, what results can be found and with what

probabilities?

2. Taking |ψ(0)〉 as the initial state, we apply a magnetic field parallel to the y axis with

strength Bo. Calculate the state **of** the system at some later time t in the {|±〉} basis.

3. Plot as a function **of** time the expectation values fo the observables Sx, Sy, and Sz. What

are the values and probabilities? Is there a relation between Bo and t for the result **of** one

**of** the measurements to be certain? Give a physical interpretation **of** this condition.

4. Again, consider the same initial state, this time at t = 0, we measure Sy and find +¯h/2

What is the state vector |ψ(0 + )〉 immediately after this measurement?

5. Now we take |ψ(0 + )〉 and apply a uniform time-dependent field parallel to the z-axis. The

Hamiltonian operator **of** the spin is then given by

H(t) = ω(t)Sz

Assume that prior to t = 0, ω(t) = 0 and for t > 0 increases linearly from 0 to ωo at time

t = T . Show that for 0 ≤ t ≤ T , the state vector can be written as

|ψ(t)〉 = 1 �

√ e

2

iθ(t) |+〉 + ie −iθ(t) |−〉 �

where θ(t) is a real function **of** t (which you need to determine).

6. Finally, at time t = τ > T , we measure Sy. What results can we find and with what

probabilities? Determine the relation which must exist between ωo and T in order for us to

be sure **of** the result. Give the physical interpretation.

179

Chapter 7

Perturbation theory

If you perturbate to much, you will go blind.

– T. A. Albright

In previous lectures, we discusses how , say through application **of** and external driving force,

the stationary states **of** a molecule or other quantum mechanical system can be come coupled

so that the system can make transitions from one state to another. We can write the transition

amplitude exactly as

G(i → j, t) = 〈j| exp(−iH(tj − ti))/¯h)|i〉 (7.1)

where H is the full Hamiltonian **of** the uncoupled system plus the applied perturbation. Thus, G

tells us the amplitude for the system prepared in state |i〉 at time ti and evolve under the applied

Hamiltonian for some time tj − ti and be found in state |j〉. In general this is a complicated

quantity to calculate. Often, the coupling is very complex. In fact, we can only exactly determine

G for a few systems: linearly driven harmonic oscillators, coupled two level systems to name the

more important ones.

In today’s lecture and following lectures, we shall develop a series **of** well defined and systematic

approximations which are widely used in all applications **of** quantum mechanics. We start

with a general solution **of** the time-independent Schrödinger equation in terms and eventually

expand the solution to infinite order. We will then look at what happens if we have a perturbation

or coupling which depends explicitly upon time and derive perhaps the most important

rule in quantum mechanics which is called: “Fermi’s Golden Rule”. 1

7.1 Perturbation Theory

In most cases, it is simply impossible to obtain the exact solution to the Schrödinger equation.

In fact, the vast majority **of** problems which are **of** physical interest can not be resolved exactly

and one is forced to make a series **of** well posed approximations. The simplest approximation is

to say that the system we want to solve looks a lot like a much simpler system which we can

1 During a seminar, the speaker mentioned Fermi’s Golden Rule. **Pr of**. Wenzel raised his arm and in German

spiked English chided the speaker that it was in fact HIS golden rule!

180

solve with some additional complexity (which hopefully is quite small). In other words we want

to be able to write our total Hamiltonian as

H = Ho + V

where Ho represents that part **of** the problem we can solve exactly and V some extra part which

we cannot. This we take as a correction or perturbation to the exact problem.

Perturbation theory can be formuated in a variery **of** ways, we begin with what is typically

termed Rayleigh-Schrödinger perturbation theory. This is the typical approach and used most

commonly. Let Ho|φn〉 = Wn|φn〉 and (Ho + λV )|ψ〉 = En|ψ〉 be the Schrödinger equations for

the uncoupled and perturbed systems. In what follows, we take λ as a small parameter and

expand the exact energy in terms **of** this parameter. Clearly, we write En as a function **of** λ and

write:

En(λ) = E (0)

n + λE (1)

n + λ 2 E (2)

n . . . (7.2)

Likewise, we can expand the exact wavefunction in terms **of** λ

|ψn〉 = |ψ (0)

n 〉 + λ|ψ (1)

n 〉 + λ 2 |ψ (2)

n 〉 . . . (7.3)

Since we require that |ψ〉 be a solution **of** the exact Hamiltonian with energy En, then

H|ψ〉 = (Ho + λV ) �

|ψ (0)

n 〉 + λ|ψ (1)

= �

E (0)

n + λE (1)

n + λ 2 E (2)

n . . .

Now, we collect terms order by order in λ

• λ 0 : Ho|ψ (0)

n 〉 = E (0)

n |ψ (0)

n 〉

• λ 1 : Ho|ψ (1)

n 〉 + V |ψ (0)

n 〉 = E (0)

n |ψ (1) 〉 + E (1)

n |ψ (0)

n 〉

n 〉 + λ 2 |ψ (2)

� �

|ψ (0)

n 〉 + λ|ψ (1)

n 〉 + λ 2 |ψ (2)

n 〉 . . . �

• λ 2 : Ho|ψ (2)

n 〉 + V |ψ (1) 〉 = E (0)

n |ψ (2)

n 〉 + E (1)

n |ψ (1)

n 〉 + E (2)

n |ψ (0)

n 〉

and so on.

n 〉 . . . �

The λ 0 problem is just the unperturbed problem we can solve. Taking the λ 1 terms and

multiplying by 〈ψ (0)

n | we obtain:

(7.4)

(7.5)

〈ψ (0)

n |Ho|ψ (0)

n 〉 + 〈ψ (0)

n |V |ψ (0) 〉 = E (0)

n 〈ψ (0)

n |ψ (1)

n 〉 + E (1)

n 〈ψ (0)

n |ψ (0)

n 〉 (7.6)

In other words, we obtain the 1st order correction for the nth eigenstate:

E (1)

n = 〈ψ (0)

n |V |ψ (0) 〉.

Note to obtain this we assumed that 〈ψ (1)

n |ψ (0)

n 〉 = 0. This is easy to check by performing a

similar calculation, except by multiplying by 〈ψ (0)

m | for m �= n and noting that 〈ψ (0)

n |ψ (0)

m 〉 = 0 are

orthogonal state.

〈ψ (0)

m |Ho|ψ (0)

n 〉 + 〈ψ (0)

m |V |ψ (0) 〉 = E (0)

n 〈ψ (0)

m |ψ (1)

n 〉 (7.7)

181

Rearranging things a bit, one obtains an expression for the overlap between the unperturbed and

perturbed states:

〈ψ (0)

m |ψ (1)

n 〉 = 〈ψ(0)

E (0)

m |V |ψ (0)

n 〉

n − E (0)

m

Now, we use the resolution **of** the identity to project the perturbed state onto the unperturbed

states:

|ψ (1)

n 〉 = �

|ψ (0)

m 〉〈ψ (0)

m |ψ (1)

n 〉

m

= �

m�=n

〈ψ (0)

m |V |ψ (0)

E (0)

n − E (0)

m

n 〉

(7.8)

|ψ (0)

m 〉 (7.9)

where we explictly exclude the n = m term to avoid the singularity. Thus, the first-order

correction to the wavefunction is

|ψn〉 ≈ |ψ (0)

n 〉 + �

This also justifies our assumption above.

m�=n

〈ψ (0)

m |V |ψ (0)

E (0)

n − E (0)

m

n 〉

|ψ (0)

m 〉. (7.10)

7.2 Two level systems subject to a perturbation

Let’s say that in the |±〉 basis our total Hamiltonian is given by

In matrix form:

H = ωSz + V Sx. (7.11)

H =

�

ω V

V −ω

Diagonalization **of** the matrix is easy, the eigenvalues are

We can also determine the eigenvectors:

where

�

(7.12)

E+ = √ ω 2 + V 2 (7.13)

E− = − √ ω 2 + V 2 (7.14)

|φ+〉 = cos(θ/2)|+〉 + sin(θ/2)|−〉 (7.15)

|φ−〉 = − sin(θ/2)|+〉 + cos(θ/2)|−〉 (7.16)

|V |

tan θ = (7.17)

ω

For constant coupling, the energy gap ω between the coupled states determines how the states

are mixed as the result **of** the coupling.

plot splitting as a function **of** unperturbed energy gap

182

7.2.1 Expansion **of** Energies in terms **of** the coupling

We can expand the exact equations for E± in terms **of** the coupling assuming that the coupling

is small compared to ω. To leading order in the coupling:

E+ = ω(1 + 1

� �

�

�|V

| �2

�

� � · · ·) (7.18)

2 � ω �

E− = ω(1 − 1

� �

�

�|V

| �2

�

� � · · ·) (7.19)

2 � ω �

On the otherhand, where the two unperturbed states are identical, we can not do this expansion

and

and

E+ = |V | (7.20)

E− = −|V | (7.21)

We can do the same trick on the wavefunctions: When ω ≪ |V | (strong coupling) , θ ≈ π/2,

Thus,

|ψ+〉 = 1

√ 2 (|+〉 + |−〉) (7.22)

|ψ−〉 = 1

√ 2 (−|+〉 + |−〉). (7.23)

In the weak coupling region, we have to first order in the coupling:

|ψ+〉 = (|+〉 +

|ψ−〉 = (|−〉 +

|V |

|−〉) (7.24)

ω

|V |

|+〉). (7.25)

ω

In other words, in the weak coupling region, the perturbed states look a lot like the unperturbed

states. Where as in the regions **of** strong mixing they are a combination **of** the unperturbed

states.

183

7.2.2 Dipole molecule in homogenous electric field

Here we take the example **of** ammonia inversion in the presence **of** an electric field. From the

problem sets, we know that the NH3 molecule can tunnel between two equivalent C3v configurations

and that as a result **of** the coupling between the two configurations, the unperturbed

energy levels Eo are split by an energy A. Defining the unperturbed states as |1〉 and |2〉 we can

define the tunneling Hamiltonian as:

or in terms **of** Pauli matrices:

H =

�

Eo −A

−A Eo

�

H = Eoσo − Aσx

Taking ψ to be the solution **of** the time-dependent Schrödinger equation

H|ψ(t)〉 = i¯h| ˙ ψ〉

we can insert the identity |1〉〈1| + |2〉〈2| = 1 and re-write this as

i¯h ˙c1 = Eoc1 − Ac2

i¯h ˙c2 = Eoc2 − Ac1

(7.26)

(7.27)

(7.28)

where c1 = 〈1|ψ〉 and c2 = 〈2|ψ〉. are the projections **of** the time-evolving wavefunction onto the

two basis states. Taking these last two equations and adding and subtracting them from each

other yields two new equations for the time-evolution:

i¯h ˙c+ = (Eo − A)c+

i¯h ˙c− = (Eo + A)c−

where c± = c1 ± c2 (we’ll normalize this later). These two new equations are easy to solve,

Thus,

�

i

c±(t) = A± exp

¯h (Eo

�

∓ A)t .

(7.29)

(7.30)

c1(t) = 1 �

eiEot/¯h A+e

2 −iAt/¯h + A−e +iAt/¯h�

and

c2(t) = 1 �

eiEot/¯h A+e

2 −iAt/¯h − A−e +iAt/¯h�

.

Now we have to specify an initial condition. Let’s take c1(0) = 1 and c2(0) = 0 corresponding to

the system starting **of**f in the |1〉 state. For this initial condition, A+ = A− = 1 and

and

c1(t) = e iEot/¯h cos(At/¯h)

c2(t) = e iEot/¯h sin(At/¯h).

184

So that the time evolution **of** the state vector is given by

|ψ(t)〉 = e iEot/¯h [|1〉 cos(At/¯h) + |2〉 sin(At/¯h)]

So, left alone, the molecule will oscillate between the two configurations at the tunneling frequency,

A/¯h.

Now, we apply an electric field. When the dipole moment **of** the molecule is aligned parallel

with the field, the molecule is in a lower energy configuration, whereas for the anti-parrallel case,

the system is in a higher energy configuration. Denote the contribution to the Hamiltonian from

the electric field as:

H ′ = µeEσz

The total Hamiltonian in the {|1〉, |2〉} basis is thus

Solving the eigenvalue problem:

we find two eigenvalues:

H =

�

Eo + µeE −A

−A Eo − µeE

λ± = Eo ±

|H − λI| = 0

�

A 2 + µ 2 eE 2 .

These are the exact eigenvalues.

In Fig. 7.1 we show the variation **of** the energy levels as a function **of** the field strength.

�

(7.31)

Figure 7.1: Variation **of** energy level splitting as a function **of** the applied field for an ammonia

molecule in an electric field

Weak field limit

If µeE/A ≪ 1, then we can use the binomial expansion

√ 1 + x 2 ≈ 1 + x 2 /2 + . . .

to write

�

A2 + µ 2 eE 2 �

= A 1 +

≈ A

�

1 + 1

2

�

µeE

A

�

µeE

A

�2 � 1

/2

�2 �

(7.32)

Thus in the weak field limit, the system can still tunnel between configurations and the energy

splitting are given by

E± ≈ (Eo ∓ A) ∓ µ2 eE 2

A

185

To understand this a bit further, let us use perturbation theory in which the tunneling

dominates and treat the external field as a perturbing force. The unperturbed hamiltonian can

be diagonalized by taking symmetric and anti-symmetric combinations **of** the |1〉 and |2〉 basis

functions. This is exactly what we did above with the time-dependent coefficients. Here the

stationary states are

|±〉 = 1

√ (|1〉 ± |2〉)

2

with energies E± = Eo ∓ A. So that in the |±〉 basis, the unperturbed Hamiltonian becomes:

H =

�

Eo − A 0

0 Eo + A

The first order correction to the ground state energy is given by

E (1) = E (0) + 〈+|H ′ |+〉

To compute 〈+|H ′ |+〉 we need to transform H ′ from the {|1〉, |2〉} uncoupled basis to the new |±〉

coupled basis. This is accomplished by inserting the identity on either side **of** H ′ and collecting

terms:

〈+|H ′ |+〉 = 〈+|(|1〉 < 1| + |2〉〈2|)H ′ (|1〉 < 1| + |2〉〈2|)〉 (7.33)

= 1

2 (〈1| + 〈2|)H′ (|1〉 + |2〉) (7.34)

= 0 (7.35)

Likewise for 〈−|H ′ |−〉 = 0. Thus, the first order correction vanish. However, since 〈+|H ′ |−〉 =

µeE does not vanish, we can use second order perturbation theory to find the energy correction.

W (2)

+ = �

H ′ miH ′ im

m�=i

Ei − Em

�

= 〈+|H′ |−〉〈−|H ′ |+〉

E (0)

+ − E (0)

−

(µeE)

=

2

Eo − A − Eo − A

= − µ2eE 2

2A

.

(7.36)

(7.37)

(7.38)

(7.39)

Similarly for W (2)

− = +µ 2 eE 2 /A. So we get the same variation as we estimated above by expanding

the exact energy levels when the field was week.

Now let us examine the wavefunctions. Remember the first order correction to the eigenstates

is given by

|+ (1) 〉 = 〈−|H′ |−〉

|−〉 (7.40)

E+ − E−

= − µE

|−〉 (7.41)

2A

186

Thus,

|+〉 = |+ (0) 〉 − µE

|−〉

2A

(7.42)

|−〉 = |− (0) 〉 + µE

|+〉

2A

(7.43)

So we see that by turning on the field, we begin to mix the two tunneling states. However, since

we have assumed that µE/A ≪ 1, the final state is not too unlike our initial tunneling states.

Strong field limit

In the strong field limit, we expand the square-root term such that � A

µeE

�

A2 + µ 2 eE 2 ⎛�

� ⎞

2 1/2

A

= Eµe ⎝ + 1⎠

µeE

⎡

= Eµe ⎣1 + 1

� � ⎤

2

A

⎦ . . .

2 µeE

≈ Eµe + 1 A

2

2

µeE

� 2

≪ 1.

(7.44)

For very strong fields, the first term dominates and the energy splitting becomes linear in the

field strength. In this limit, the tunneling has been effectively suppressed.

Let us analyze this limit using perturbation theory. Here we will work in the |1, 2〉 basis and

treat the tunneling as a perturbation. Since the electric field part **of** the Hamiltonian is diagonal

in the 1,2 basis, our unperturbed strong-field hamiltonian is simply

H =

�

Eo − µeE 0

0 Eo − µeE

�

(7.45)

and the perturbation is the tunneling component. As before, the first-order corrections to the

energy vanish and we are forced to resort to 2nd order perturbation theory to get the lowest

order energy correction. The results are

W (2) = ± A2

2µE

which is exactly what we obtained by expanding the exact eigenenergies above. Likewise, the

lowest-order correction to the state-vectors are

|1〉 = |1 0 〉 − A

2µE |20 〉 (7.46)

|2〉 = |2 0 〉 + A

2µE |10 〉 (7.47)

So, for large E the second order correction to the energy vanishes, the correction to the wavefunction

vanishes and we are left with the unperturbed (i.e. non-tunneling) states.

187

7.3 Dyson Expansion **of** the Schrödinger Equation

The Rayleigh-Schrödinger approach is useful for discrete spectra. However, it is not very useful

for scattering or systems with continuous spectra. On the otherhand, the Dyson expansion **of** the

wavefunction can be applied to both cases. Its development is similar to the Rayleigh-Schrödinger

case, We begin by writing the Schrödinger Equation as usual:

(Ho + V )|ψ〉 = E|ψ〉 (7.48)

where we define |φ〉 and W to be the eigenvectors and eigenvalues **of** part **of** the full problem.

We shall call this the “uncoupled” problem and assume it is something we can easily solve.

Ho|φ〉 = W |φ〉 (7.49)

We want to write the solution **of** the fully coupled problem in terms **of** the solution **of** the

uncoupled problem. First we note that

(Ho − E)|ψ〉 = V |ψ〉. (7.50)

Using the “uncoupled problem” as a “homogeneous” solution and the coupling as an inhomogeneous

term, we can solve the Schrödinger equation and obtain |ψ〉 EXACTLY as

This may seem a bit circular. But we can iterate the solution:

Or, out to all orders:

1

|ψ〉 = |φ〉 + V |ψ〉 (7.51)

Ho − E

1

1

|ψ〉 = |φ〉 + V |φ〉 +

Ho − E Ho − E V

1

V |ψ〉. (7.52)

Ho − W

|ψ〉 = |φ〉 +

∞�

�

1

Ho − E V

�n

|φ〉 (7.53)

n=1

Assuming that the series converges rapidly (true for V

i.e.

Likewise:

where

|ψ (1)

n 〉 = |φn〉 + �

�

|ψ (2)

n 〉 = |ψ (1)

n 〉 + �

�

lm

n

1

Wn − Wm

1

(Wm − Wl)(Wn − Wm)

�

|φm〉〈φn|V |φm〉 (7.58)

� 2

VlmVmn|φn〉 (7.59)

Vlm = 〈φl|V |φm〉 (7.60)

is the matrix element **of** the coupling in the uncoupled basis. These last two expressions are the

first and second order corrections to the wavefunction.

Note two things. First that I can actually solve the perturbation series exactly by noting that

the series has the form **of** a geometric progression, for x < 1 converge uniformly to

Thus, I can write

1

1 − x = 1 + x + x2 + · · · =

|ψ〉 =

=

=

n=0

∞�

∞�

x

n=0

n

(7.61)

∞�

�

1

Ho − E V

�n

|φ〉 (7.62)

n=0

(GoV ) n |φ〉 (7.63)

1

|φ〉 (7.64)

1 − GoV

where Go = (Ho−E) −1 (This is the “time-independent” form **of** the propagator for the uncoupled

system). This particular analysis is particularly powerful in deriving the propagator for the fully

coupled problem.

We now calculate the first order and second order corrections to the energy **of** the system.

To do so, we make use **of** the wavefunctions we just derived and write

E (1)

n = 〈ψ (0)

n |H|ψ (0)

n 〉 = Wn + 〈φn|V |φn〉 = Wn + Vnn (7.65)

So the lowest order correction to the energy is simply the matrix element **of** the perturbation

in the uncoupled or unperturbed basis. That was easy. What about the next order correction.

Same procedure as before: (assuming the states are normalized)

E (2)

n = 〈ψ (1)

n |H|ψ (1)

n 〉

= 〈φn|H|φn〉

+ �

�

〈φn|H|φm〉

m�=n

= Wn + Vnn + �

1

Wn − Wm

|Vnm| 2

m�=n

Wn − Wm

189

�

〈φm|V |φn〉 + O[V 3 ]

(7.66)

Notice that I am avoiding the case where m = n as that would cause the denominator to

be zero leading to an infinity. This must be avoided. The so called “degenerate case” must

be handled via explicit matrix diagonalization. Closed forms can be obtained for the doubly

degenerate case easily.

Also note that the successive approximations to the energy require one less level **of** approximation

to the wavefunction. Thus, second-order energy corrections are obtained from first order

wavefunctions.

7.4 Van der Waals forces

7.4.1 Origin **of** long-ranged attractions between atoms and molecules

One **of** the underlyuing principles in chemistry is that molecules at long range are attractive

towards each other. This is clearly true for polar and oppositely charges species. It is also true

for non-polar and neutral species, such as methane, noble gases, and etc. These forces are due to

polarization forces or van der Waals forces, which is is attractive and decreases as 1/R 7 , i.e. the

attractive part **of** the potential goes as −1/R 6 . In this section we will use perturbation theory

to understand the origins **of** this force, restricting our attention to the interaction between two

hydrogen atoms separated by some distance R.

Let us take the two atoms to be motionless and separated by distance R with �n being the

vector pointing from atom A to atom B. Now let �ra be the vector connecting nuclei A to its

electron and likewise for �rB. Thus each atom has an instantaneous electric dipole moment

�µa = q � Ra

(7.67)

�µb = q � Rb. (7.68)

(7.69)

We will assume that R ≫ ra & rb so that the electronic orbitals on each atom do not come into

contact.

Atom A creates an electrostatic potential, U, for atom B in which the charges in B can

interact. This creates an interaction energy W . Since both atoms are neutral, the most important

source for the interactions will come from the dipole-dipole interactions. This, the dipole **of** A

interacts with an electric field E = −∇U generated by the dipole field about B and vice versa. To

calculate the dipole-dipole interaction, we start with the expression for the electrostatic potential

created by µa at B.

U(R) = 1 µa · R

4πɛo R3 Thus,

�E = −∇U = − q 1

4πɛo R3 (�ra − 3(�ra · �n)�n) .

Thus the dipole-dipole interaction energy is

W = −�µb · � E

= e2

R 3 (�ra · �rb − 3(�ra · �n)(�rb · �n)) (7.70)

190

where e 2 = q 2 /4πɛo. Now, let’s set the z axis to be along �n so we can write

W = e2

R 3 (xaxb + yayb − 2zazb).

This will be our perturbing potential which we add to the total Hamiltonian:

H = Ha + Hb + W

where Ha are the unperturbed Hamiltonians for the atoms. Let’s take for example wo hydrogens

each in the 1s state. The unperturbed system has energy

H|1s1; 1s2〉 = (E1 + E2)|1s1; 1s2〉 = −2EI|1s1; 1s2〉,

where EI is the ionization energy **of** the hydrogen 1s state (EI = 13.6eV). The first order vanishes

since it involves integrals over odd functions. This we can anticipate since the 1s orbitals are

spatially isotropic, so the time averaged valie **of** the dipole moments is zero. So, we have to look

towards second order corrections.

The second order energy correction is

E (2) = �

nlm

�

n ′ l ′ m ′ = |〈nlm; n′ l ′ m ′ |W |1sa; 1sb〉| 2

−2EI − En − En ′

where we restrict the summation to avoid the |1sa; 1ab〉 state. Since W ∝ 1/R 3 and the deminator

is negative, we can write

E (2) = − C

R 6

which explains the origin **of** the 1/R 6 attraction.

Now we evaluate the proportionality constant C Written explicitly,

�

4

C = e

�

nml n ′ l ′ m ′

|〈nlm ′ n ′ l ′ m ′ |(xaxb + yayb − 2zazb)|1sa; 1sb〉| 2

2EI + En + En ′

(7.71)

Since n and n ′ ≥ 2 and |En| = EI/n2 < EI, we can replace En and En ′ with 0 with out

appreciable error. Now, we can use the resolution **of** the identity

1 = �

to remove the summation and we get

C = e4

2EI

�

nml n ′ l ′ m ′

|nlm; n ′ l ′ m ′ 〉〈nlm; n ′ l ′ m ′ |

〈1sa; 1ab|(xaxb + yayb − 2zazb) 2 |1sa; 1sb〉 (7.72)

where EI is the ionization potential **of** the 1s state (EI = 1/2). Surprisingly, this is simple

to evaluate since we can use symmetry to our advantage. Since the 1s orbitals are spherically

symmetric, any term involving cross-terms **of** the sort

〈1sa|xaya|1s〉 = 0

191

vanish. This leaves only terms **of** the sort

〈1s|x 2 |1s〉.

all **of** which are equal to 1/3 **of** the mean value **of** RA = x 2 a + y 2 a + z 2 a. Thus,

where ao is the Bohr radius. Thus,

C = 6 e2

�

�

�

�

2EI

〈1s|R

3 |1s〉

�

�

�

�

2

= 6e 2 ao

E (2) 2 ao

= −6e

R6 What does all this mean. We stated at the beginning that the average dipole moment **of** a H

1s atom is zero. That does not mean that every single measurement **of** µa will yield zero. What is

means is that the probability **of** finding the atom with a dipole moment µa is the same for finding

the dipole vector pointed in the opposite direction. Adding the two together produces a net zero

dipole moment. So its the fluctuations about the mean which give the atom an instantaneous

dipole field. Moreover, the fluctuations in A are independent **of** the fluctuations in B, so first

order effects must be zero since the average interaction is zero.

Just because the fluctuations are independent does not mean they are not correlated. Consider

the field generated by A as felt by B. This field is due to the fluctuating dipole at A. This field

induces a dipole at B. This dipole field is in turn felt by A. As a result the fluctuations become

correlated and explains why this is a second order effect. In a sense, A interacts with is own

dipole field through “reflection” **of**f B.

7.4.2 Attraction between an atom a conducting surface

The interaction between an atom or molecule and a surface is a fundimental physical process

in surface chemistry. In this example, we will use perturbation theory to understand the longranged

attraction between an atom, again taking a H 1s atom as our species for simplicity, and

a conducting surface. We will take the z axis to be normal to the surface and assume that the

atom is high enough **of**f the surface that its altitude is much larger than atomic dimensions.

Furthermore, we will assume that the surface is a metal conductor and we will ignore any atomic

level **of** detail in the surface. Consequently, the atom can only interact with its dipole image on

the opposite side **of** the surface.

We can use the same dipole-dipole interaction as before with the following substitutions

e 2 −→ −e 2

(7.73)

R −→ 2d (7.74)

xb −→ x ′ a = xa (7.75)

yb −→ y ′ a = ya (7.76)

zb −→ z ′ a = −za (7.77)

where the sign change reflects the sign difference in the image charges. So we get

W = − e2

8d 3 (x2 a + y 2 a + 2z 2 a)

192

as the interaction between a dipole and its image. Taking the atom to be in the 1s ground state,

the first order term is non-zero:

E (1) = 〈1s|W |1s〉.

Again, using spherical symmetry to our advantage:

E (1) = − e2

8d3 4〈1s|r2 |1s〉 = − e2a2 o

2d

Thus an atom is attracted to the wall with an interaction energy which varries as 1/d 3 . This is

a first order effect since there is perfect correlation between the two dipoles.

7.5 Perturbations Acting over a Finite amount **of** Time

Perhaps the most important application **of** perturbation theory is in cases in which the coupling

acts for a finite amount **of** time. Such as the coupling **of** a molecule to a laser field. The laser field

impinges upon a molecule at some instant in time and some time later is turned **of**f. Alternatively

we can consider cases in which the perturbation is slowly ramped up from 0 to a final value.

7.5.1 General form **of** time-dependent perturbation theory

In general, we can find the time-evolution **of** the coefficients by solving

i¯h ˙cn(t) = Encn(t) + �

λWnk(t)ck(t) (7.78)

where λWnk are the matrix elements **of** the perturbation. Now, let’s write the cn(t) as

cn(t) = bn(t)e −iEnt

and assume that bn(t) changes slowly in time. Thus, we can write a set **of** new equations for the

bn(t) as

k

3 .

i¯h ˙ bn = �

e iωnkt

λWnkbk(t) (7.79)

Now we assume that the bn(t) can be written as a perturbation expansion

k

bn(t) = b (0)

n (t) + λb (1)

n (t) + λ 2 b (2)

n (t) + · · ·

where as before λ is some dimensionless number **of** order unity. Taking its time derivative and

equating powers **of** λ one finds

and that b (0)

n (t) = 0.

i¯h ˙ b (i)

n = �

e iωnkt

Wnk(t)b (i−1)

k

k

193

Now, we calculate the first order solution. For t < 0 the system is assumed to be in some

well defined initial state, |φi〉. Thus, only one bn(t < 0) coefficient can be non-zero and must be

independent **of** time since the coupling has not been turned on. Thus,

bn(t = 0) = δni

At t = 0, we turn on the coupling and λW jumps from 0 to λW (0). This must hold for all order

in λ. So, we immediately get

b (0)

n (0) = δni (7.80)

b (i)

n (0) = 0 (7.81)

Consequently, for all t > 0, b (0)

n (t) = δni which completely specifies the zeroth order result. This

also gives us the first order result.

which is simple to integrate

i¯h ˙ b (1)

n (t) = �

b (1)

n (t) = − i

k

e iωnkt Wnkδki

e

¯h 0

iωnis

Wni(s)ds.

Thus, our perturbation wavefunction is written as

|ψ(t)〉 = e −iEot/¯h |φo〉 + �

7.5.2 Fermi’s Golden Rule

= e iωnit Wni(t) (7.82)

� t

n�=0

λb (1)

n (t)e −iEnt/¯h |φn〉 (7.83)

Let’s consider the time evolution **of** a state under a small perturbation which varies very slowly

in time. The expansion coefficients **of** the state in some basis **of** eigenstates **of** the unperturbed

Hamiltonian evolve according to:

i¯h ˙cs(t) = �

Hsncn(t) (7.84)

where Hsn is the matrix element **of** the full Hamiltonian in the basis.

n

Hsn = Esδns + Vsn(t) (7.85)

Assuming that V (t) is slowly varying and that Vsn

= Escs(t) + �

VsnAn(t)e −i/¯hEnt

n

(7.89)

We now proceed to solve this equation via series **of** well defined approximations. Our first

assumption is that Vsn

= 4|〈s| ˆ V |n〉| 2

� �

2 sin ((Es − En − ¯hω)t/(2¯h))

. (7.102)

(Es − En − ¯hω) 2

This is the general form for a harmonic perturbation. The function

sin 2 (ax)

x 2

(7.103)

is called the sinc function or the “Mexican Hat Function”. At x = 0 the peak is sharply peaked

(corresponding to Es − En − ¯hω = 0). Thus, the transition is only significant when the energy

difference matches the frequency **of** the applied perturbation. As t → ∞ (a = t/(2¯h)), the peak

become very sharp and becomes a δ-function. The width **of** the peak is

∆x = 2π¯h

t

(7.104)

Thus, the longer we measure a transition, the more well resolved the measurement will become.

This has pr**of**ound implications for making measurements **of** meta-stable systems.

We have an expression for the transition probability between two discrete states. We have

not taken into account the the fact that there may be more than one state close by nor have we

taken into account the finite “width” **of** the perturbation (instrument function). When there a

many states close the the transition, I must take into account the density **of** nearby states. Thus,

we define

ρ(E) = ∂N(E)

∂E

(7.105)

as the density **of** states close to energy E where N(E) is the number **of** states with energy E.

Thus, the transition probability to any state other than my original state is

Ps(t) = �

Pns(t) = �

|As(t)| 2

to go from a discrete sum to an integral, we replace

Thus,

�

→

n

s

� �

dN

dE =

dE

s

(7.106)

ρ(E)dE (7.107)

�

Ps(t) = dE|As(t)| 2 ρ(E) (7.108)

Since |As(t)| 2 is peaked sharply at Es = Ei +¯hω we can treat ρ(E) and ˆ Vsn as constant and write

Taking the limits **of** integration to ±∞

Ps(t) = 4|〈s| ˆ V |n〉| 2 � 2 sin (ax)

ρ(Es)

x2 dx (7.109)

� ∞

−∞

dx sin2 (ax)

x

196

= πa = πt

2¯h

(7.110)

In other words:

We can also define a transition rate as

Ps(t) = 2πt

¯h |〈s| ˆ V |n〉| 2 ρ(Es) (7.111)

R(t) =

P (t)

t

thus, the Golden Rule Transition Rate from state n to state s is

(7.112)

Rn→s(t) = 2π

¯h |〈s| ˆ V |n〉| 2 ρ(Es) (7.113)

This is perhaps one **of** the most important approximations in quantum mechanics in that it

has implications and applications in all areas, especially spectroscopy and other applications **of**

matter interacting with electro-magnetic radiation.

7.6 Interaction between an atom and light

What I am going to tell you about is what we teach our physics students in the third

or fourth year **of** graduate school... It is my task to convince you not to turn away

because you don’t understand it. You see my physics students don’t understand it...

That is because I don’t understand it. Nobody does.

Richard P. Feynman, QED, The Strange Theory **of** Light and Matter

Here we explore the basis **of** spectroscopy. We will consider how an atom interacts with a

photon field in the low intensity limit in which dipole interactions are important. We will then

examine non-resonant excitation and discuss the concept **of** oscillator strength. Finally we will

look at resonant emission and absorption concluding with a discussion **of** spontaneous emission.

In the next section, we will look at non-linear interactions.

7.6.1 Fields and potentials **of** a light wave

An electromagnetic wave consists **of** two oscillating vector field components which are perpendicular

to each other and oscillate at at an angular frequency ω = ck where k is the magnitude

**of** the wavevector which points in the direction **of** propagation and c is the speed **of** light. For

such a wave, we can always set the scalar part **of** its potential to zero with a suitable choice in

gauge and describe the fields associate with the wave in terms **of** a vector potential, � A given by

�A(r, t) = Aoeze iky−iωt + A ∗ oeze −iky+iωt

Here, the wave-vector points in the +y direction, the electric field, E is polarized in the yz plane

and the magnetic field B is in the xy plane. Using Maxwell’s relations

�E(r, t) = − ∂A

∂t = iωez(Aoe i(ky−ωt) − A ∗ oe −i(ky−ωt) )

197

and

�B(r, t) = ∇ × � A = ikex(Aoe i(ky−ωt) − A ∗ oe −i(ky−ωt) ).

We are free to choose the time origin, so we will choos it as to make Ao purely imaginary and

set

where E and B are real quantities such that

Thus

iωAo = E/2 (7.114)

ikAo = B/2 (7.115)

E

B

= ω

k

= c.

E(r, t) = Eez cos(ky − ωt) (7.116)

B(r, t) = Bez sin(ky − ωt) (7.117)

where E and B are the magnitudes **of** the electric and magnetic field components **of** the plane

wave.

Lastly, we define what is known as the Poynting vector (yes, it’s pronounced pointing) which

is parallel to the direction **of** propagation:

�S = ɛoc 2 � E × � B. (7.118)

Using the expressions for � E and � B above and averaging over several oscillation periods:

�S

2 E

= ɛoc

2 ey

7.6.2 Interactions at Low Light Intensity

(7.119)

The electromagnetic wave we just discussed can interact with an atomic electron. The Hamiltonian

**of** this electron can be given by

H = 1

2m (P − qA(r, t))2 + V (r) − q

S · B(r, t)

m

where the first term represents the interaction between the electron and the electrical field **of**

the wave and the last term represents the interaction between the magnetic moment **of** the

electron and the magnetic moment **of** the wave. In expanding the kinetic energy term, we have

to remember that momentum and position do not commute. However, in the present case, A is

parallel to the z axis and Pz and y commute. So, we wind up with the following:

where

H = Ho + W

Ho = P2

+ V (r)

2m

198

is the unperturbed (atomic) hamiltonian and

W = − q q q2

P · A − S · B +

m m 2m A2 .

The first two depend linearly upon A and the second is quadratic in A. So, for low intensity we

can take

W = − q q

P · A − S · B.

m m

Before moving on, we will evaluate the relative importance **of** each term by orders **of** magnitude

for transitions between bound states. In the second term, the contribution **of** the spin operator

is on the order **of** ¯h and the contribution from B is on the order **of** kA. Thus,

WB

WE

=

q

m

q

m

S · B ¯hk

≈

P · A p

¯h/p is on the order **of** an atomic radius, ao and k = 2π/λ where λ is the wavelength **of** the light,

typically on the order **of** 1000ao. Thus,

WB

WE

≈ ao

λ

So, the magnetic coupling is not at all important and we focus only upon the coupling to the

electric field.

Using the expressions we derived previously, the coupling to the electric field component **of**

the light wave is given by:

Now, we expand the exponential in powers **of** y

≪ 1.

WE = − q

m pz(Aoe iky e −iωt + A ∗ oe −iky e +iωt ).

e ±iky = 1 ± iky − 1

2 k2 y 2 + . . .

since ky ≈ ao/λ ≪ 1, we can to a good approximation keep only the first term. Thus we get the

dipole operator

WD = qE

mω pz sin(ωt).

In the electric dipole approximation, W (t) = WD(t).

Note, that one might expect that WD should have been written as

WD = −qEz cos(ωt)

since we are, after all, talking about a dipole moment associated with the motion **of** the electron

about the nucleus. Actually, the two expressions are identical! The reason is that I can always

choose a differnent gauge to represent the physical problem without changing the physical result.

To get the present result, we used

A = E

ω ez sin(ωt)

199

and

U(r) = 0

as the scalar potential. A gauge transformation is introduced by taking a function, f and defining

a new vector potential and a new scalar potential as

A ′ = A + ∇f

U ′ = U − ∂f

∂t

We are free to choose f however we do so desire. Let’s take f = zE sin(ωt)/ω. Thus,

and

A ′ E

= ez (sin(ky − ωt) + sin(ωt))

ω

U ′ = −zE cos ωt

is the new scalar potential. In the electric dipole approximation, ky is small, so we set ky = 0

everywhere and obtain A ′ = 0. Thus, the total Hamiltonian becomes

with perturbation

H = Ho + qU ′ (r, t)

W ′ D = −qzE cos(ωt).

This is the usual form **of** the dipole coupling operator. However, when we do the gauge transformation,

we have to transform the state vector as well.

Next, let us consider the matrix elements **of** the dipole operator between two stationary states

**of** Ho: |ψi〉 and |ψf〉 with eigenenergy Ei and Ef respectively. The matrix elements **of** WD are

given by

Wfi(t) = qE

mω sin(ωt)〈ψf|pz|ψi〉

We can evaluate this by noting that

Thus,

[z, Ho] = i¯h ∂Ho

∂pz

= i¯h pz

m .

〈ψf|pz|ψi〉 = imωfi〈ψf|z|ψi〉.

Consequently,

sin(ωt)

Wfi(t) = iqEωfi zfi.

ω

Thus, the matrix elements **of** the dipole operator are those **of** the position operator. This determines

the selection rules for the transition.

Before going through any specific details, let us consider what happens if the frequency ω

does not coincide with ωfi. Soecifically, we limit ourselves to transitions originating from the

ground state **of** the system, |ψo〉. We will assume that the field is weak and that in the field the

200

atom acquires a time-dependent dipole moment which oscillates at the same frequency as the

field via a forced oscillation. To simplify matters, let’s assume that the electron is harmonically

bound to the nucleus in a classical potential

V (r) = 1 2

mωor

where ωo is the natural frequency **of** the electron.

The classical motion **of** the electron is given by the equations **of** motion (via the Ehrenfest

theorem)

¨z + ω 2 z = qE

m cos(ωt).

This is the equation **of** motion for a harmonic oscillator subject to a periodic force. This inhomogeneous

differential equation can be solved (using Fourier transform methods) and the result

is

z(t) = A cos(ωot − φ) +

2

qE

m(ω 2 o − ω 2 ) cos(ωt)

where the first term represents the harmonic motion **of** the electron in the absence **of** the driving

force. The two coefficients, A and φ are determined by the initial condition. If we have a very

slight damping **of** the natural motion, the first term dissappears after a while leaving only the

second, forced oscillation, so we write

z =

qE

m(ω 2 o − ω 2 ) cos(ωt).

Thus, we can write the classical induced electric dipole moment **of** the atom in the field as

D = qz =

q 2 E

m(ω 2 o − ω 2 ) cos(ωt).

Typically this is written in terms **of** a susceptibility, χ, where

χ =

q 2

m(ω 2 o − ω 2 ) .

Now we look at this from a quantum mechanical point **of** view. Again, take the initial state

to be the ground state and H = Ho + WD as the Hamiltonian. Since the time-evolved state can

be written as a superposition **of** eigenstates **of** Ho,

|ψ(t)〉 = �

cn(t)|φn〉

n

To evaluate this we can us the results derived previously in our derivation **of** the golden rule,

|ψ(t)〉 = |φo〉 + � qE

n�=0 2im¯hω 〈n|pz|φo〉

×

�

−iωnot iωt

e − e

ωno + ω − e−iωnot − e−iωt �

|φn〉

ωno − ω

(7.120)

where we have removed a common phase factor. We can then calculate the dipole moment

expectation value, 〈D(t)〉 as

〈D(t)〉 = 2q2 � ωon|〈φn|z|φo〉|

E cos(ωt)

¯h n

2

ω2 no − ω2 201

(7.121)

Oscillator Strength

We can now notice the similarity between a driven harmonic oscillator and the expectation value

**of** the dipole moment **of** an atom in an electric field. We can define the oscillator strength as a

dimensionless and real number characterizing the transition between |φo and |φn〉

fno = 2mωno

|〈φn|z|φo〉|

¯h

2

In Exercise 2.4, we proved the Thomas-Reiche-Kuhn sum rule, which we can write in terms **of**

the oscillator strengths, �

fno = 1

This can be written in a very compact form:

n

m

¯h 2 〈φo|[x, [H, x]]|φo〉 = 1.

7.6.3 Photoionization **of** Hydrogen 1s

Up until now we have considered transitions between discrete states inrtoduced via some external

perturbation. Here we consider the single photon photoionization **of** the hydrogen 1s orbital to

illustrate how the golden rule formalism can be used to calculate photoionization cross-sections

as a function **of** the photon-frequency. We already have an expression for dipole coupling:

WD = qE

mω pz sin(ωt) (7.122)

and we have derived the golden rule rate for transitions between states:

Rif = 2π

¯h |〈f|V |i〉|2 δ(Ei − Ef + ¯hω). (7.123)

For transitions to the continuum, the final states are the plane-waves.

ψ(k) = 1

Ω 1/2 eik·r . (7.124)

where Ω is the volume element. Thus the matrix element 〈1s|V |k〉 can be written as

〈1s|pz|k〉 = ¯hkz

Ω1/2 �

ψ1s(r)e ik·r dr. (7.125)

To evaluate the integral, we need to transform the plane-wave function in to spherical coordinates.

This can be done vie the expansion;

e ik·r = �

i l (2l + 1)jl(kr)Pl(cos(θ)) (7.126)

l

where jl(kr) is the spherical Bessel function and Pl(x) is a Legendre polynomial, which we can

also write as a spherical harmonic function,

�

4π

Pl(cos(θ)) =

2l + 1 Yl0(θ, φ). (7.127)

202

Thus, the integral we need to perform is

〈1s|k〉 = 1

√ πΩ

�

��

l

Y ∗

�

00Yl0dΩ i l�

� ∞

4π(2l + 1) r

0

2 e −r jl(kr)dr. (7.128)

The angular integral we do by orthogonality and produces a delta-function which restricts the

sum to l = 0 only leaving,

The radial integral can be easily performed using

leaving

Thus, the matrix element is given by

〈1s|k〉 = 1

� ∞

√ r

Ω 0

2 e −r j0(kr)dr. (7.129)

j0(kr) = sin(kr)

kr

〈1s|k〉 = 4

k

1

Ω 1/2

〈1s|V |k〉 = qE¯h

mω

(7.130)

1

(1 + k2 . (7.131)

) 2

1

Ω1/2 2

(1 + k 2 ) 2

(7.132)

This we can indert directly in to the golden rule formula to get the photoionization rate to a

given k-state.

R0k = 2π¯h

Ω

� �2

qE

mω

which we can manipulate into reading as

R0k = 16π

¯hΩ

4

(1 + k 2 ) 4 δ(Eo − Ek + ¯hω). (7.133)

� �2

qE

m

mω

δ(k2 − K2 )

(1 + k2 ) 4

(7.134)

where we write K 2 = 2m(EI + ¯hω)/¯h 2 to make our notation a bit more compact. Eventually,

we want to know the rate as a function **of** the photon frequency, so let’s put everything except

the frequency and the volume element into a single constant, I, which is related to the intensity

**of** the incident photon.

R0k = I 1

Ω ω2 δ(k2 − K2 )

(1 + k2 . (7.135)

) 4

Now, we sum over all possible final states to get the total photoionization rate. To do this, we

need to turn the sum over final states into an integral, this is done by

�

k

= Ω

� ∞

4π k

(2π) 3

0

2 dk (7.136)

203

Thus,

R = I 1

Ω ω2 �

Ω ∞

4π

(2π) 3

0

= I

ω2 1

2π2 � ∞

0

k 2 δ(k2 − K2 )

(1 + k2 dk

) 4

k 2 δ(k2 − K2 )

(1 + k2 dk

) 2

Now we do a change **of** variables: y = k 2 and dy = 2kdk so that the integral become

� ∞

0

k 2 δ(k2 − K2 )

(1 + k2 �

1 ∞

dk =

) 2 2 0

=

y 1/2

K

2(1 + K 2 ) 4

(1 + y 2 ) 4 δ(y − K2 )dy

Pulling everything together, we see that the total photoionization rate is given by

R = I

ω2 1

2π2 K

(1 + K 2 ) 4

I

=

� m

¯h 2

√

ω ¯h − ɛo

√ �

2 π2 ω2 1 +

√

2 ω − 1

= I

32 π 2 ω 6

2 m (ω ¯h−ɛo)

¯h 2

�4 (7.137)

(7.138)

where in the last line we have converted to atomic units to clean things up a bit. This expression

is clearly valid only when ¯hω > EI = 1/2hartree (13.6 eV) and a plot **of** the photo-ionization

rate is given in Fig. 7.2

7.6.4 Spontaneous Emission **of** Light

The emission and absorption **of** light by an atom or molecule is perhaps the most spectacular

and important phenomena in the universe. It happens when an atom or molecule undergoes a

transition from one state to another due to its interaction with the electro-magnetic field. Because

the electron-magnetic field can not be entirely eliminated from any so called isolated system

(except for certain quantum confinement experiments), no atom or molecule is ever really isolated.

Thus, even in the absence **of** an explicitly applied field, an excited system can spontaneously emit

a photon and relax to a lower energy state. Since we have all done spectroscopy experiments

at one point in our education or another, we all know that the transitions are between discrete

energy levels. In fact, it was in the examination **of** light passing through glass and light emitted

from flames that people in the 19th century began to speculate that atoms can absorb and emit

light only at specific wavelengths.

We will use the GR to deduce the probability **of** a transition under the influence **of** an applied

light field (laser, or otherwise). We will argue that the system is in equilibrium with the electromagnetic

field and that the laser drive