Graduate Quantum Mechanics Class Notes Leon Hostetler May 4, 2019 Version 0.0 Contents Preface v 1 The Origins of Quantum Mechanics1 1.1 Introduction................................... 1 1.2 Bohr Quantization............................... 2 1.3 Virial Theorem................................. 5 1.4 Hierarchy of Lengths.............................. 10 1.5 The Correspondence Principle......................... 10 1.6 Simple Variational Approximations...................... 12 1.7 Summary: The Origins of Quantum Mechanics............... 16 2 Wave Mechanics 19 2.1 Infinite Walls.................................. 19 2.2 Continuity Equation.............................. 22 2.3 Finite Walls................................... 23 2.4 Propagating Waves............................... 28 2.5 Resonance.................................... 29 2.6 Tunneling Waves................................ 30 2.7 Level Density.................................. 32 2.8 Summary: Wave Mechanics.......................... 38 3 Wave Functions and Operators 41 3.1 Superposition Principle ............................ 41 3.2 Dirac Delta Function.............................. 41 3.3 Representations of the Wave Function .................... 43 3.4 Operators.................................... 45 3.5 Green's Functions ............................... 49 3.6 Transformations ................................ 50 3.7 Summary: Wave Functions and Operators.................. 53 4 Formalism 56 4.1 Groups...................................... 56 4.2 Hilbert Space.................................. 57 4.3 Hermitian Operators.............................. 59 4.4 Commutators.................................. 62 4.5 Uncertainty................................... 65 4.6 Summary: Formalism ............................. 68 5 Quantum Dynamics 70 5.1 Unitary Transformations............................ 70 5.2 Operator Dynamics and the Heisenberg Picture............... 72 5.3 Conservation Laws............................... 77 5.4 Continuity Equation.............................. 78 Contents iii 5.5 Wave Packets.................................. 79 5.6 Decay of States................................. 81 5.7 Survival Probability .............................. 83 5.8 Time Reversal.................................. 84 5.9 Summary: Quantum Dynamics........................ 86 6 General 1D Problems 88 6.1 General Properties............................... 88 6.2 1D Harmonic Oscillator ............................ 90 6.3 Multidimensional Harmonic Oscillators.................... 94 6.4 Molecular Excitations ............................. 95 6.5 Secondary Quantization............................ 96 6.6 Heisenberg-Weyl Algebra ........................... 97 6.7 Coherent States................................. 100 6.8 Summary: General 1D Problems ....................... 104 7 Magnetic Field 106 7.1 Minimal Coupling (Spinless Particles) .................... 106 7.2 Particle in a Magnetic Field.......................... 107 7.3 Gauge Invariance................................ 110 7.4 Landau Levels ................................. 111 7.5 Aharonov-Bohm Effect............................. 112 7.6 Summary: Magnetic Field........................... 114 8 3D Motion 116 8.1 Angular Momentum Algebra ......................... 116 8.2 Orbital Momentum and Spherical Functions . 120 8.3 Schrodinger Equation in Spherical Coordinates . 122 8.4 Hydrogen Atom................................. 123 8.5 Symmetries and Conservation Laws...................... 128 8.6 Summary: 3D Motion ............................. 131 9 Spin and Angular Momentum Coupling 134 9.1 Spin-1/2..................................... 134 9.2 Angular Momentum Coupling......................... 137 9.3 Atomic Splitting ................................ 142 9.4 Wigner-Eckart Theorem............................ 145 9.5 Electromagnetic Multipoles .......................... 147 9.6 Summary: Spin and Angular Momentum Coupling . 150 10 Many-body Quantum Mechanics 153 10.1 Identical Particles ............................... 153 10.2 Gibbs' Paradox................................. 154 10.3 Transposition Operator ............................ 155 10.4 Bosons and Fermions.............................. 156 10.5 Second Quantization.............................. 158 10.6 Nuclear Configurations............................. 161 10.7 Atomic Configurations............................. 164 10.8 Summary: Many-body Quantum Mechanics . 169 11 Approximation Methods 173 11.1 Variational Method............................... 173 11.2 The Two-State System............................. 174 11.3 Diagonalization on Restricted Basis...................... 175 iv Contents 11.4 Stationary Perturbation Theory........................ 178 11.5 Time-dependent Perturbation Theory .................... 182 11.6 Periodic Perturbations............................. 184 11.7 Summary: Approximation Methods...................... 187 12 Atoms 189 12.1 Lamb Shift ................................... 189 12.2 Polarizability .................................. 192 12.3 Van der Waals Forces ............................. 196 12.4 Summary: Atoms................................ 198 13 Photons and the Electromagnetic Field 199 13.1 Hamiltonian of EM Field ........................... 199 13.2 Quantization of the EM Field......................... 201 13.3 Radiation.................................... 203 13.4 Electric Dipole Transitions........................... 205 13.5 Multipole Transitions ............................. 210 13.6 Photoabsorption ................................ 212 13.7 Summary: Photons and the EM Field .................... 214 14 Relativistic Quantum Mechanics 216 14.1 Klein-Gordon Equation ............................ 216 14.2 Four-vectors and Minkowski Space...................... 220 14.3 Dirac Equation................................. 222 14.4 Relativistic Corrections ............................ 235 14.5 Summary: Relativistic Quantum Mechanics . 240 15 Scattering 243 15.1 Perturbation Theory.............................. 243 15.2 Potential Scattering .............................. 245 15.3 Born (Series) Approximation ......................... 248 15.4 The Method of Partial Waves......................... 252 15.5 Scattering of Particles with Spin ....................... 259 15.6 Scattering of Identical Particles........................ 263 15.7 Summary: Scattering.............................. 264 Preface About These Notes My class notes can be found at www.leonhostetler.com/classnotes These are my class notes from a two-semester graduate course (PHY 851-852) on quantum mechanics taken at Michigan State University. The course was taught by Profes- sor Vladimir Zelevinsky. The primary textbooks used were \Quantum Physics" Volumes I and II both by Professor Zelevinsky. Please bear in mind that these notes will contain errors. Any errors are certainly my own. If you find one, please email me at [email protected] with the name of the class notes, the page on which the error is found, and the nature of the error. This work is currently licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License. That means you are free to copy and distribute this document in whole for noncommercial use, but you are not allowed to distribute derivatives of this document or to copy and distribute it for commercial reasons. Updates Last Updated: May 4, 2019 Version 0.5 (May 4, 2019): Initial upload Chapter 1 The Origins of Quantum Mechanics In this text, we will primarily use Gaussian units. In these units, 4π0 ! 1. 1.1 Introduction Planck and the Blackbody Distribution In classical mechanics, the action is the integral of the Lagrangian L dt along the trajectory taken by a particle. Planck's constant h is the quantum of´ action, and has value h = 6:626 × 10−34 Js = 6:626 × 10−27 ergs: Planck introduced the quantum of action in 1900. He was interested in solving the problem of finding the distribution of the frequencies of radiation in a box due to temperature, if the box was in equilibrium. Classically, this resulted in several paradoxes. Instead of writing energy in terms of hν it is often more convenient to write it in terms of ~!, where h = ; ~ 2π is the reduced Planck constant. Einstein and the Photoelectric Effect The photoelectric effect is the effect in which shining a light on a metal causes electrons to be ejected from the metal. The effect was explained by Einstein in 1905. There is a minimum energy, called a work function W or ionization energy that must be added to electrons in a metal to free the electron from the atom. The work function is specific to the material. Consider an electron in a metal. If E < 0, then the electron is bound to an atom. If E > 0, the electron is unbounded and undergoes free motion. At E = 0, we have the threshold energy. Consider the diagram shown below. To eject an electron from a metal, we need to add some energy W (i.e. the work function) to bring the bound electron's energy up to the threshold. We need to add additional energy to tear the electron loose from the atom. This additional energy is the kinetic energy of the electron which is now a free particle. 2 The Origins of Quantum Mechanics Paradoxically, increasing the intensity of the light does not increase the (kinetic) energy of the kicked off photons. Rather it simply increases the number of electrons that get ejected. As the plot below illustrates, the kinetic energy of the ejected photons depends on the frequency (i.e. energy) of the incident light, and not on the intensity