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

Recommendations

Info

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
Page 1 and 2: Quantum Mechanics Lecture Notes for
Page 3 and 4: Contents 0 Introduction 8 0.1 Essen
Page 5: 5.3.1 Harmonic Oscillators and Nucl
Page 9 and 10: List of Tables 3.1 Location of node
Page 11 and 12: exactly on paper and move on to dis
Page 13 and 14: - What is Quantum Mechanics?, Trans
Page 15 and 16: 0.3 2003 Course Calendar This is a
Page 17 and 18: Chapter 1 Survey of Classical Mecha
Page 19 and 20: Taking time derivatives ˙x(t) =
Page 21 and 22: Since δ ˙x = dδx/dt and integrat
Page 23 and 24: The equations of motion are d ∂L
Page 25 and 26: This example also illustrates anoth
Page 27 and 28: Our goal is to write this Hamiltoni
Page 29 and 30: Now, let’s take a time average of
Page 31 and 32: Chapter 2 Waves and Wavefunctions I
Page 33 and 34: We can also use the |φ(k)〉 = |k
Page 35 and 36: 0.75 0.5 0.25 -3 -2 -1 1 2 3 -0.25
Page 37 and 38: Figure 2.3: Go for fixed t as a fun
Page 39 and 40: Since ψ(x) must vanish at x = 0 an
Page 41 and 42: We will choose the coefficients c1
Page 43 and 44: symêasym 4 3 2 1 -1 -2 -3 -4 2 4 6
Page 45 and 46: 1 0.95 T 0.9 0.85 10 En 20 30 -10 4
Page 47 and 48: Im@yD 4 2 0 -2 -4 -10 x 0 10 -5 0 F
Page 49 and 50: 30 20 10 DOS Quantum well vs. 3d bo
Page 51 and 52: 12 10 5 -5 -10 -15 -20 -25 8 6 4 2
Page 53 and 54: Exercise 2.4 Consider a particle wi
Page 55 and 56: Exercise 2.8 Consider the Hamiltoni
Page 57 and 58:
Chapter 3 Semi-Classical Quantum Me
Page 59 and 60:
for circular orbits, then the theor
Page 61 and 62:
Substituting Eq. 3.17 into χ = ¯h
Page 63 and 64:
If we make the approximation that
Page 65 and 66:
where a and b are the turning point
Page 67 and 68:
This gives us the transmission prob
Page 69 and 70:
and the WKB wavefunction becomes:
Page 71 and 72:
15 10 5 -5 -10 2 4 6 8 10 Figure 3.
Page 73 and 74:
Thus, where we use the fact that y
Page 75 and 76:
Thus, once we know χ(b) for a give
Page 77 and 78:
Exercise 3.5 The impact parameter,
Page 79 and 80:
We transform this into an integral
Page 81 and 82:
1. Assume that ψo and ψ1 are solu
Page 83 and 84:
Figure 4.1: Gaussian distribution f
Page 85 and 86:
Figure 4.3: Constructive and destru
Page 87 and 88:
4.0.1 The description of a physical
Page 89 and 90:
4.0.5 The Superposition Principle L
Page 91 and 92:
Similarly for P2. Part of our job i
Page 93 and 94:
Thus, Thus, Let’s see if ˆx and
Page 95 and 96:
Now, we reinsert the definitions of
Page 97 and 98:
Therefore, and 〈µ|a〉 = � 〈
Page 99 and 100:
The matrix A is Hermitian if A = A
Page 101 and 102:
Thus, we can show that F ( Â)|φa
Page 103 and 104:
4.3 Bohr Frequency and Selection Ru
Page 105 and 106:
We can also compute: 〈p 2 〉 =
Page 107 and 108:
ack to the ground state or some low
Page 109 and 110:
Solution: You can either do this th
Page 111 and 112:
Exercise 4.8 For this section, cons
Page 113 and 114:
For Domain 1 we have: For Domain 2
Page 115 and 116:
and ∂y ′ ∂α = ∂η ∂x we
Page 117 and 118:
the pivot point. In any case, the g
Page 119 and 120:
5.2.3 Variational method applied to
Page 121 and 122:
Thus, we can expand any state |ψ
Page 123 and 124:
φo(x) = φo(x) = 1 φo(x) = � φ
Page 125 and 126:
Since What happends if I do the sam
Page 127 and 128:
eigenstate withe energy En − ¯h
Page 129 and 130:
〈φm|p|φn〉 = i � mω¯h 2
Page 131 and 132:
Since we know that a † acting on
Page 133 and 134:
5.3.2 Classical interpretation. In
Page 135 and 136:
Figure 5.4: Quantum and Classical P
Page 137 and 138:
Figure 5.5: London-Eyring-Polanyi-S
Page 139 and 140:
Figure 5.7: Model potential for pro
Page 141 and 142:
5.4.2 Numerical Diagonalization A m
Page 143 and 144:
Table 5.1 lists a the first few of
Page 145 and 146:
V[x_] := a*(x^4 - x^2); cmm = 8064*
Page 147 and 148:
1. Express the total Hamiltonian as
Page 149 and 150:
Figure 5.10: Ammonia Inversion and
Page 151 and 152:
(d) Since the ψL and ψR basis fun
Page 153 and 154:
Bibliography [1] C. A. Parr and D.
Page 155 and 156:
6.1 Quantum Theory of Rotations Let
Page 157 and 158:
Which we can summarize as = i¯hLz
Page 159 and 160:
Thus, L 2 = − 1 � 1 ∂ sin θ
Page 161 and 162:
6.4 Eigenfunctions of L 2 The wavef
Page 163 and 164:
For the case of m = 0, Yl0 = i l
Page 165 and 166:
We can use this to write � Y ∗
Page 167 and 168:
i.e. (x = cos θ) and satisfy Pl(x)
Page 169 and 170:
When l is very large the sin 2 ((l
Page 171 and 172:
Table 6.2: Relation between various
Page 173 and 174:
To solve this equation, we first ha
Page 175 and 176:
The proceedure is to substitute thi
Page 177 and 178:
with and The closure, or idempotent
Page 179 and 180:
6.10 Problems and Exercises Exercis
Page 181 and 182:
2. Express the stationary states as
Page 183 and 184:
solve with some additional complexi
Page 185 and 186:
7.2.1 Expansion of Energies in term
Page 187 and 188:
So that the time evolution of the s
Page 189 and 190:
Thus, |+〉 = |+ (0) 〉 − µE |
Page 191 and 192:
i.e. Likewise: where |ψ (1) n 〉
Page 193 and 194:
where e 2 = q 2 /4πɛo. Now, let
Page 195 and 196:
as the interaction between a dipole
Page 197 and 198:
= Escs(t) + � VsnAn(t)e −i/¯hE
Page 199 and 200:
In other words: We can also define
Page 201 and 202:
is the unperturbed (atomic) hamilto
Page 203 and 204:
atom acquires a time-dependent dipo
Page 205 and 206:
Thus, the integral we need to perfo
Page 207 and 208:
R HarbL 0.8 0.6 0.4 0.2 0.5 1 1.5 2
Page 209 and 210:
The number of atoms going from 2 to
Page 211 and 212:
= 4 ω 3 3 ¯hc3 e2 r 4πɛo 2 . (7
Page 213 and 214:
other. The tunneling rate can be es
Page 215 and 216:
where 1F1(a, b, z) is the Hypergeom
Page 217 and 218:
Integrating over the electronic deg
Page 219 and 220:
3. At t = 0, the initial classical
Page 221 and 222:
where b = � (Fj(0) − Fi(0)) 2
Page 223 and 224:
energy level for R = 1fm and compar
Page 225 and 226:
These states obey analogous rules f
Page 227 and 228:
where the columns represent the par
Page 229 and 230:
The remaining two must be construct
Page 231 and 232:
Not too bad, the actual result is
Page 233 and 234:
the J integral for the 1s1s configu
Page 235 and 236:
where the sum is over the nuclei an
Page 237 and 238:
where Sij is the overlap between th
Page 239 and 240:
Exercise 8.2 Verify the expressions
Page 241 and 242:
Appendix: Creation and Annihilation
Page 243 and 244:
8.5 Problems and Exercises Exercise
Page 245 and 246:
Figure 8.4: Setup calculation dialo
Page 247 and 248:
Figure 8.5: HOMO-1, HOMO and LUMO f
Page 249 and 250:
Appendix A Physical Constants and C
Page 251 and 252:
Appendix B Mathematical Results and
Page 253 and 254:
Figure B.1: sin(xa)/πx representat
Page 255 and 256:
B.2 Coordinate systems In each case
Page 257 and 258:
B.2.3 Cylindrical x φ • Coordina
Page 259:
Ground-state Configuration Protacti
show all

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

Create successful ePaper yourself

Delete template?

Save as template?