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

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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 (offered 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
Page 1 and 2: Quantum Mechanics Lecture Notes for
Page 3 and 4: Contents 0 Introduction 8 0.1 Essen
Page 5 and 6: 5.3.1 Harmonic Oscillators and Nucl
Page 7 and 8: List of Figures 1.1 Tangent field f
Page 9: List of Tables 3.1 Location of node
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
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Substituting Eq. 3.17 into χ = ¯h
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If we make the approximation that
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where a and b are the turning point
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This gives us the transmission prob
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and the WKB wavefunction becomes:
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15 10 5 -5 -10 2 4 6 8 10 Figure 3.
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Thus, where we use the fact that y
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Thus, once we know χ(b) for a give
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Exercise 3.5 The impact parameter,
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We transform this into an integral
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1. Assume that ψo and ψ1 are solu
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Figure 4.1: Gaussian distribution f
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Figure 4.3: Constructive and destru
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4.0.1 The description of a physical
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4.0.5 The Superposition Principle L
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Similarly for P2. Part of our job i
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Thus, Thus, Let’s see if ˆx and
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Now, we reinsert the definitions of
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Therefore, and 〈µ|a〉 = � 〈
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The matrix A is Hermitian if A = A
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Thus, we can show that F ( Â)|φa
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4.3 Bohr Frequency and Selection Ru
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We can also compute: 〈p 2 〉 =
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ack to the ground state or some low
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Solution: You can either do this th
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Exercise 4.8 For this section, cons
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For Domain 1 we have: For Domain 2
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and ∂y ′ ∂α = ∂η ∂x we
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the pivot point. In any case, the g
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5.2.3 Variational method applied to
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Thus, we can expand any state |ψ
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φo(x) = φo(x) = 1 φo(x) = � φ
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Since What happends if I do the sam
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eigenstate withe energy En − ¯h
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〈φm|p|φn〉 = i � mω¯h 2
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Since we know that a † acting on
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5.3.2 Classical interpretation. In
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Figure 5.4: Quantum and Classical P
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Figure 5.5: London-Eyring-Polanyi-S
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Figure 5.7: Model potential for pro
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5.4.2 Numerical Diagonalization A m
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Table 5.1 lists a the first few of
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V[x_] := a*(x^4 - x^2); cmm = 8064*
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1. Express the total Hamiltonian as
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Figure 5.10: Ammonia Inversion and
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(d) Since the ψL and ψR basis fun
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Bibliography [1] C. A. Parr and D.
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6.1 Quantum Theory of Rotations Let
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Which we can summarize as = i¯hLz
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Thus, L 2 = − 1 � 1 ∂ sin θ
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6.4 Eigenfunctions of L 2 The wavef
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For the case of m = 0, Yl0 = i l
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We can use this to write � Y ∗
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i.e. (x = cos θ) and satisfy Pl(x)
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When l is very large the sin 2 ((l
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Table 6.2: Relation between various
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To solve this equation, we first ha
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The proceedure is to substitute thi
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with and The closure, or idempotent
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6.10 Problems and Exercises Exercis
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2. Express the stationary states as
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solve with some additional complexi
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7.2.1 Expansion of Energies in term
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So that the time evolution of the s
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Thus, |+〉 = |+ (0) 〉 − µE |
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i.e. Likewise: where |ψ (1) n 〉
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where e 2 = q 2 /4πɛo. Now, let
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as the interaction between a dipole
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= Escs(t) + � VsnAn(t)e −i/¯hE
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In other words: We can also define
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is the unperturbed (atomic) hamilto
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atom acquires a time-dependent dipo
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Thus, the integral we need to perfo
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R HarbL 0.8 0.6 0.4 0.2 0.5 1 1.5 2
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The number of atoms going from 2 to
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= 4 ω 3 3 ¯hc3 e2 r 4πɛo 2 . (7
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other. The tunneling rate can be es
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where 1F1(a, b, z) is the Hypergeom
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Integrating over the electronic deg
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3. At t = 0, the initial classical
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where b = � (Fj(0) − Fi(0)) 2
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energy level for R = 1fm and compar
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These states obey analogous rules f
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where the columns represent the par
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The remaining two must be construct
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Not too bad, the actual result is
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the J integral for the 1s1s configu
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where the sum is over the nuclei an
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where Sij is the overlap between th
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Exercise 8.2 Verify the expressions
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Appendix: Creation and Annihilation
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8.5 Problems and Exercises Exercise
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Figure 8.4: Setup calculation dialo
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Figure 8.5: HOMO-1, HOMO and LUMO f
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Appendix A Physical Constants and C
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Appendix B Mathematical Results and
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Figure B.1: sin(xa)/πx representat
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B.2 Coordinate systems In each case
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B.2.3 Cylindrical x φ • Coordina
Page 259:
Ground-state Configuration Protacti
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Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

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