Quantum Mechanics Made Simple: Lecture Notes Weng Cho CHEW1 September 23, 2013 1The author is with U of Illinois, Urbana-Champaign. He works part time at Hong Kong U this summer. Contents Preface vii Acknowledgements vii 1 Introduction 1 1.1 Introduction . .1 1.2 Quantum Mechanics is Bizarre . .2 1.3 The Wave Nature of a Particle{Wave Particle Duality . .2 2 Classical Mechanics and Some Mathematical Preliminaries 5 2.1 Introduction . .5 2.2 Lagrangian Formulation . .6 2.3 Hamiltonian Formulation . .9 2.4 More on Hamiltonian . 10 2.5 Poisson Bracket . 11 2.6 Some Useful Knowledge of Matrix Algebra . 11 2.6.1 Identity, Hermitian, Symmetric, Inverse and Unitary Matrices . 13 2.6.2 Determinant . 14 2.6.3 Eigenvectors and Eigenvalues . 14 2.6.4 Trace of a Matrix . 15 2.6.5 Function of a Matrix . 15 3 Quantum Mechanics|Some Preliminaries 19 3.1 Introduction . 19 3.2 Probabilistic Interpretation of the Wavefunction . 20 3.3 Time Evolution of the Hamiltonian Operator . 21 3.4 Simple Examples of Time Independent Schr¨odingerEquation . 24 3.4.1 Particle in a 1D Box . 24 3.4.2 Particle Scattering by a Barrier . 25 3.4.3 Particle in a Potential Well . 26 3.5 The Quantum Harmonic Oscillator{A Preview . 28 i ii Quantum Mechanics Made Simple 4 Time-Dependent Schr¨odingerEquation 31 4.1 Introduction . 31 4.2 Quantum States in the Time Domain . 31 4.3 Coherent State . 32 4.4 Measurement Hypothesis and Expectation Value . 33 4.4.1 Uncertainty Principle{A Simple Version . 36 4.4.2 Particle Current . 37 5 More Mathematical Preliminaries 39 5.1 A Function is a Vector . 39 5.2 Operators . 42 5.2.1 Matrix Representation of an Operator . 42 5.2.2 Bilinear Expansion of an Operator . 43 5.2.3 Trace of an Operator . 44 5.2.4 Unitary Operators . 45 5.2.5 Hermitian Operators . 46 5.3 *Identity Operator in a Continuum Space . 49 5.4 *Changing Between Representations . 52 5.4.1 Momentum Operator . 52 5.4.2 Position Operator . 53 5.4.3 The Coordinate Basis Function . 54 5.5 Commutation of Operators . 54 5.6 Expectation Value and Eigenvalue of Operators . 55 5.7 *Generalized Uncertainty Principle . 57 5.8 *Time Evolution of the Expectation Value of an Operator . 59 5.9 Periodic Boundary Condition . 60 6 Approximate Methods in Quantum Mechanics 63 6.1 Introduction . 63 6.2 Use of an Approximate Subspace . 63 6.3 *Time Independent Perturbation Theory . 65 6.3.1 First Order Perturbation . 68 6.3.2 Second Order Perturbation . 69 6.3.3 Higher Order Corrections . 69 6.4 Tight Binding Model . 70 6.4.1 Variational Method . 73 6.5 Time Dependent Perturbation Theory . 74 7 Quantum Mechanics in Crystals 77 7.1 Introduction . 77 7.2 Bloch-Floquet Waves . 78 7.2.1 Periodicity of E(k)............................. 80 7.2.2 Symmetry of E(k) with respect to k ................... 80 7.3 Bloch-Floquet Theorem for 3D . 80 Contents iii 7.4 Fermi-Dirac Distribution Function . 83 7.4.1 Semiconductor, Metal, and Insulator . 84 7.4.2 Why Do Electrons and Holes Conduct Electricity? . 86 7.5 Effective Mass Schr¨odingerEquation . 86 7.6 Heterojunctions and Quantum Wells . 88 7.7 Density of States (DOS) . 89 7.7.1 Fermi Level and Fermi Energy . 90 7.7.2 DOS in a Quantum Well . 91 7.7.3 Quantum Wires . 93 8 Angular Momentum 97 8.1 Introduction . 97 8.1.1 Electron Trapped in a Pill Box . 98 8.1.2 Electron Trapped in a Spherical Box . 100 8.2 Mathematics of Angular Momentum . 103 8.2.1 Transforming to Spherical Coordinates . 104 9 Spin 109 9.1 Introduction . 109 9.2 Spin Operators . 109 9.3 The Bloch Sphere . 112 9.4 Spinor . 112 9.5 Pauli Equation . 113 9.5.1 Splitting of Degenerate Energy Level . 115 9.6 Spintronics . 115 10 Identical Particles 121 10.1 Introduction . 121 10.2 Pauli Exclusion Principle . 122 10.3 Exchange Energy . 123 10.4 Extension to More Than Two Particles . 124 10.5 Counting the Number of Basis States . 126 10.6 Examples . 127 10.7 Thermal Distribution Functions . 128 11 Density Matrix 131 11.1 Pure and Mixed States . 131 11.2 Density Operator . 132 11.3 Time Evolution of the Matrix Element of an Operator . 135 11.4 Two-Level Quantum Systems . 136 11.4.1 Interaction of Light with Two-Level Systems . 137 iv Quantum Mechanics Made Simple 12 Quantization of Classical Fields 147 12.1 Introduction . 147 12.2 The Quantum Harmonic Oscillator Revisited . 148 12.2.1 Eigenfunction by the Ladder Approach . 150 12.3 Quantization of Waves on a Linear Atomic Chain{Phonons . 151 12.4 Schr¨odingerPicture versus Heisenberg Picture . 156 12.5 The Continuum Limit . 158 12.6 Quantization of Electromagnetic Field . 160 12.6.1 Hamiltonian . 162 12.6.2 Field Operators . 163 12.6.3 Multimode Case and Fock State . 164 12.6.4 One-Photon State . 165 12.6.5 Coherent State Revisited . 166 12.7 Thermal Light . 170 13 Schr¨odingerWave Fields 173 13.1 Introduction . 173 13.2 Fock Space for Fermions . 173 13.3 Field Operators . 175 13.4 Similarity Transform . 177 13.5 Additive One-Particle Operator . 178 13.5.1 Three-Particle Case . 180 13.6 Additive Two-Particle Operator . ..