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- K2.chem.uh.edu

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

- Page 2 and 3: Lecture Notes on Quantum Chemistry
- Page 4 and 5: 3 Semi-Classical Quantum Mechanics
- Page 6 and 7: 7.8 Problems and Exercises . . . .
- Page 8 and 9: 3.5 Form of the radial wave for rep
- Page 10 and 11: Chapter 0 Introduction Nothing conv
- Page 12 and 13: 0.1 Essentials • Instructor: Prof
- Page 14 and 15: 4. These are the only problems you
- Page 16 and 17: Part I Lecture Notes 14
- Page 18 and 19: 1.1 Newton’s equations of motion
- Page 20 and 21: superimposed curve is one trajector
- Page 22 and 23: You may be wondering at this point
- Page 24 and 25: Z' Y Z Y' where we notice that −p
- Page 26 and 27: So, we can take the quantity in the
- Page 28 and 29: 1.4.2 Time dependence of a dynamica
- Page 30 and 31: Figure 1.3: Screen shot of using Ma
- Page 32 and 33: The position operator acts on the s
- Page 34 and 35: then Since for any operator Thus, w
- Page 36 and 37: 2.2.2 Evolution of ψ(x) Now, let
- Page 38 and 39: Figure 2.4: Evolution of a free par
- Page 40 and 41: 14 12 10 8 6 4 2 -1 1 2 3 4 Figure
- Page 42 and 43: or for the symmetric case and � f
- Page 44 and 45: 1 0.8 0.6 0.4 0.2 R,T 10 20 30 40 E
- Page 46 and 47: 1.5 1 0.5 -10 -5 5 10 -0.5 -1 -1.5
- Page 48 and 49: 3 2.5 2 1.5 1 0.5 DOS which gives 0
- Page 50 and 51: 1 0.8 0.6 0.4 0.2 -0.2 j lHxL 5 10
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hydrogen atom, etc...). We will ret

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1. f(x) = e −x 2. f(x) = e −x2

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Exercise 2.11 Consider a particle o

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3.1 Bohr-Sommerfield quantization L

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3.2 The WKB Approximation The origi

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Each of these terms involves E −

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If we stick to regions where the se

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The particular case we are interest

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Absorbing the coefficient into a ne

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node xn 1 -2.33811 2 -4.08795 3 -5.

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3.4 Scattering b r θ Figure 3.4: E

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we know the asymptotic velocity, v,

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3.4.3 Quantum treatment The quantum

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1 0.5 -0.5 -1 5 10 15 20 25 30 Figu

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for θ(l + 1/2) ≫ 1. Thus, we can

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Chapter 4 Postulates of Quantum Mec

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Figure 4.2: Combination of two dist

- Page 86 and 87:
2. Those that go through slit 2. We

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4.0.3 Quantum Measurement: The only

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This operator is called the “idem

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4.0.7 The temporal evolution of the

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For example, the expectation value

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4.1 Dirac Notation and Linear Algeb

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So in short; thus S † = (S T )

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1. [ Â, ˆ B] = −[ ˆ B, Â] 2.

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Exercise 4.3 Consider the following

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Figure 4.4: The diffraction functio

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The phase factor is relatively unim

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Finally, according to the time-ener

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Exercise 4.7 1. Let |φn〉 be the

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Chapter 5 Bound States of The Schr

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5.2 The Variational Principle Often

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So, the only way for this to be tru

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Figure 5.1: Variational paths betwe

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5.2.4 Variational theorems: Rayleig

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To evaluate this, we break the prob

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The X and P obey the canonical comu

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After cleaning things up: iP φo(x)

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1. [N, a] = [a † a, a] = −a 2.

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Figure 5.2: Hermite Polynomials, Hn

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Consequently, the hermite polynomia

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1 0.8 0.6 0.4 0.2 -4 -2 2 4 2 1 -4

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5.3.3 Molecular Vibrations The full

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V HkCalêmolL 600 400 200 -200 -400

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1 0.5 -2 -1 1 2 -0.5 -1 -2 -1 1 2 F

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1 0.75 0.5 0.25 -1 -0.5 0.5 1 -0.25

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The way we use this is to use the

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5.5 Problems and Exercises Exercise

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1. What boundary conditions must ea

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(b) We first will consider the barr

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2. the total energy is fixed � ni

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Chapter 6 Quantum Mechanics in 3D I

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must be at least proportional to th

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Figure 6.1: Vector model for the qu

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(note: we used [Lz, L±] = ±L± )

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Table 6.1: Spherical Harmonics (Con

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6.5 Addition theorem and matrix ele

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From tables, So Thus, � C l+1,0 l

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and θ = 0,cooresponding to the cas

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Figure 6.3: Classical and Quantum P

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6.8 Motion in a central potential:

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Finally, we have to normalize R �

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a neutral paramagnetic Ag atom in i

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Where B is the magnitude of the fie

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Exercise 6.8 The σx matrix is give

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Chapter 7 Perturbation theory If yo

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Rearranging things a bit, one obtai

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7.2.2 Dipole molecule in homogenous

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To understand this a bit further, l

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7.3 Dyson Expansion of the Schrödi

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Notice that I am avoiding the case

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vanish. This leaves only terms of t

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Now, we calculate the first order s

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= 4|〈s| ˆ V |n〉| 2 � � 2 s

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and �B(r, t) = ∇ × � A = ike

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and U(r) = 0 as the scalar potentia

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Oscillator Strength We can now noti

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Thus, R = I 1 Ω ω2 � Ω ∞ 4

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where �µ is the dipole moment ve

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= ω3 |〈2|�µ · �ɛ|1〉|2

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Under the BO approximation, the nuc

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(T ) where ϕ n (x) is the coordina

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where xs is the classical turning p

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So, J(t) represents the time-depend

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a slightly different phase contribu

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V(hartree) -20 -40 -60 -80 -100 -12

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Chapter 8 Many Body Quantum Mechani

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While these symmetry requirement ar

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However, we could have also written

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since φα(x) is an eigenstate of H

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We now look for a a set of orbitals

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• Method development: The develop

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Cleaning things up, 〈Ψ e m(R(t))

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S,J,A 1 0.8 0.6 0.4 0.2 Figure 8.1:

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e+,e- HhartreeL 0.25 0.2 0.15 0.1 0

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We can prove similar results for th

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Exercise 8.4 In this problem we con

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states undergo the largest change u

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Figure 8.6: Transition state geomet

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Table A.1: Physical Constants Const

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B.1.2 Properties Some useful proper

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Here the height is proportional to

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B.2.2 Spherical • Coordinates:

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Appendix C Mathematica Notebook Pag