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

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

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

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
  • Page 52 and 53:

    hydrogen atom, etc...). We will ret

  • Page 54 and 55:

    1. f(x) = e −x 2. f(x) = e −x2

  • Page 56 and 57:

    Exercise 2.11 Consider a particle o

  • Page 58 and 59:

    3.1 Bohr-Sommerfield quantization L

  • Page 60 and 61:

    3.2 The WKB Approximation The origi

  • Page 62 and 63:

    Each of these terms involves E −

  • Page 64 and 65:

    If we stick to regions where the se

  • Page 66 and 67:

    The particular case we are interest

  • Page 68 and 69:

    Absorbing the coefficient into a ne

  • Page 70 and 71:

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

  • Page 72 and 73:

    3.4 Scattering b r θ Figure 3.4: E

  • Page 74 and 75:

    we know the asymptotic velocity, v,

  • Page 76 and 77:

    3.4.3 Quantum treatment The quantum

  • Page 78 and 79:

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

  • Page 80 and 81:

    for θ(l + 1/2) ≫ 1. Thus, we can

  • Page 82 and 83:

    Chapter 4 Postulates of Quantum Mec

  • Page 84 and 85:

    Figure 4.2: Combination of two dist

  • Page 86 and 87:

    2. Those that go through slit 2. We

  • Page 88 and 89:

    4.0.3 Quantum Measurement: The only

  • Page 90 and 91:

    This operator is called the “idem

  • Page 92 and 93:

    4.0.7 The temporal evolution of the

  • Page 94 and 95:

    For example, the expectation value

  • Page 96 and 97:

    4.1 Dirac Notation and Linear Algeb

  • Page 98 and 99:

    So in short; thus S † = (S T )

  • Page 100 and 101:

    1. [ Â, ˆ B] = −[ ˆ B, Â] 2.

  • Page 102 and 103:

    Exercise 4.3 Consider the following

  • Page 104 and 105:

    Figure 4.4: The diffraction functio

  • Page 106 and 107:

    The phase factor is relatively unim

  • Page 108 and 109:

    Finally, according to the time-ener

  • Page 110 and 111:

    Exercise 4.7 1. Let |φn〉 be the

  • Page 112 and 113:

    Chapter 5 Bound States of The Schr

  • Page 114 and 115:

    5.2 The Variational Principle Often

  • Page 116 and 117:

    So, the only way for this to be tru

  • Page 118 and 119:

    Figure 5.1: Variational paths betwe

  • Page 120 and 121:

    5.2.4 Variational theorems: Rayleig

  • Page 122 and 123:

    To evaluate this, we break the prob

  • Page 124 and 125:

    The X and P obey the canonical comu

  • Page 126 and 127:

    After cleaning things up: iP φo(x)

  • Page 128 and 129:

    1. [N, a] = [a † a, a] = −a 2.

  • Page 130 and 131:

    Figure 5.2: Hermite Polynomials, Hn

  • Page 132 and 133:

    Consequently, the hermite polynomia

  • Page 134 and 135:

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

  • Page 136 and 137:

    5.3.3 Molecular Vibrations The full

  • Page 138 and 139:

    V HkCalêmolL 600 400 200 -200 -400

  • Page 140 and 141:

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

  • Page 142 and 143:

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

  • Page 144 and 145:

    The way we use this is to use the

  • Page 146 and 147:

    5.5 Problems and Exercises Exercise

  • Page 148 and 149:

    1. What boundary conditions must ea

  • Page 150 and 151:

    (b) We first will consider the barr

  • Page 152 and 153:

    2. the total energy is fixed � ni

  • Page 154 and 155:

    Chapter 6 Quantum Mechanics in 3D I

  • Page 156 and 157:

    must be at least proportional to th

  • Page 158 and 159:

    Figure 6.1: Vector model for the qu

  • Page 160 and 161:

    (note: we used [Lz, L±] = ±L± )

  • Page 162 and 163:

    Table 6.1: Spherical Harmonics (Con

  • Page 164 and 165:

    6.5 Addition theorem and matrix ele

  • Page 166 and 167:

    From tables, So Thus, � C l+1,0 l

  • Page 168 and 169:

    and θ = 0,cooresponding to the cas

  • Page 170 and 171:

    Figure 6.3: Classical and Quantum P

  • Page 172 and 173:

    6.8 Motion in a central potential:

  • Page 174 and 175:

    Finally, we have to normalize R �

  • Page 176 and 177:

    a neutral paramagnetic Ag atom in i

  • Page 178 and 179:

    Where B is the magnitude of the fie

  • Page 180 and 181:

    Exercise 6.8 The σx matrix is give

  • Page 182 and 183:

    Chapter 7 Perturbation theory If yo

  • Page 184 and 185:

    Rearranging things a bit, one obtai

  • Page 186 and 187:

    7.2.2 Dipole molecule in homogenous

  • Page 188 and 189:

    To understand this a bit further, l

  • Page 190 and 191:

    7.3 Dyson Expansion of the Schrödi

  • Page 192 and 193:

    Notice that I am avoiding the case

  • Page 194 and 195:

    vanish. This leaves only terms of t

  • Page 196 and 197:

    Now, we calculate the first order s

  • Page 198 and 199:

    = 4|〈s| ˆ V |n〉| 2 � � 2 s

  • Page 200 and 201:

    and �B(r, t) = ∇ × � A = ike

  • Page 202 and 203:

    and U(r) = 0 as the scalar potentia

  • Page 204 and 205:

    Oscillator Strength We can now noti

  • Page 206 and 207:

    Thus, R = I 1 Ω ω2 � Ω ∞ 4

  • Page 208 and 209:

    where �µ is the dipole moment ve

  • Page 210 and 211:

    = ω3 |〈2|�µ · �ɛ|1〉|2

  • Page 212 and 213:

    Under the BO approximation, the nuc

  • Page 214 and 215:

    (T ) where ϕ n (x) is the coordina

  • Page 216 and 217:

    where xs is the classical turning p

  • Page 218 and 219:

    So, J(t) represents the time-depend

  • Page 220 and 221:

    a slightly different phase contribu

  • Page 222 and 223:

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

  • Page 224 and 225:

    Chapter 8 Many Body Quantum Mechani

  • Page 226 and 227:

    While these symmetry requirement ar

  • Page 228 and 229:

    However, we could have also written

  • Page 230 and 231:

    since φα(x) is an eigenstate of H

  • Page 232 and 233:

    We now look for a a set of orbitals

  • Page 234 and 235:

    • Method development: The develop

  • Page 236 and 237:

    Cleaning things up, 〈Ψ e m(R(t))

  • Page 238 and 239:

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

  • Page 240 and 241:

    e+,e- HhartreeL 0.25 0.2 0.15 0.1 0

  • Page 242 and 243:

    We can prove similar results for th

  • Page 244 and 245:

    Exercise 8.4 In this problem we con

  • Page 246 and 247:

    states undergo the largest change u

  • Page 248 and 249:

    Figure 8.6: Transition state geomet

  • Page 250 and 251:

    Table A.1: Physical Constants Const

  • Page 252 and 253:

    B.1.2 Properties Some useful proper

  • Page 254 and 255:

    Here the height is proportional to

  • Page 256 and 257:

    B.2.2 Spherical • Coordinates:

  • Page 258 and 259:

    Appendix C Mathematica Notebook Pag

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