Quantum Mechanics Made Simple: Lecture Notes Weng Cho CHEW 1 June 2, 2015 1The author is with U of Illinois, Urbana-Champaign. 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 2.2.1 The Classical Harmonic Oscillator . .9 2.2.2 Continuum Mechanics of a String . .9 2.3 Hamiltonian Formulation . 11 2.4 More on Hamiltonian . 13 2.5 Poisson Bracket . 13 2.6 Some Useful Knowledge of Matrix Algebra . 14 2.6.1 Identity, Hermitian, Symmetric, Inverse and Unitary Matrices . 15 2.6.2 Determinant . 16 2.6.3 Eigenvectors and Eigenvalues . 17 2.6.4 Trace of a Matrix . 17 2.6.5 Function of a Matrix . 18 3 Quantum Mechanics|Some Preliminaries 21 3.1 Introduction . 21 3.2 Probabilistic Interpretation of the Wavefunction . 22 3.3 Time Evolution of the Hamiltonian Operator . 23 3.4 Simple Examples of Time Independent Schr¨odingerEquation . 26 3.4.1 Particle in a 1D Box . 26 3.4.2 Particle Scattering by a Barrier . 27 3.4.3 Particle in a Potential Well . 28 3.5 The Quantum Harmonic Oscillator{A Preview . 30 i ii Quantum Mechanics Made Simple 4 Time-Dependent Schr¨odingerEquation 33 4.1 Introduction . 33 4.2 Quantum States in the Time Domain . 33 4.3 Coherent State . 34 4.4 Measurement Hypothesis and Expectation Value . 35 4.4.1 Uncertainty Principle{A Simple Version . 38 4.4.2 Particle Current . 39 5 More Mathematical Preliminaries 41 5.1 A Function is a Vector . 41 5.2 Operators . 44 5.2.1 Matrix Representation of an Operator . 44 5.2.2 Bilinear Expansion of an Operator . 45 5.2.3 Trace of an Operator . 46 5.2.4 Unitary Operators . 47 5.2.5 Hermitian Operators . 48 5.3 *Identity Operator in a Continuum Space . 51 5.4 *Changing Between Representations . 54 5.4.1 Momentum Operator . 54 5.4.2 Position Operator . 55 5.4.3 The Coordinate Basis Function . 56 5.5 Commutation of Operators . 56 5.6 Expectation Value and Eigenvalue of Operators . 57 5.7 *Generalized Uncertainty Principle . 59 5.8 *Time Evolution of the Expectation Value of an Operator . 61 5.8.1 Comparison to classical equations of motion . 62 5.9 Periodic Boundary Condition . 64 6 Approximate Methods in Quantum Mechanics 67 6.1 Introduction . 67 6.2 Use of an Approximate Subspace . 67 6.3 *Time Independent Perturbation Theory . 69 6.3.1 First Order Perturbation . 72 6.3.2 Second Order Perturbation . 73 6.3.3 Higher Order Corrections . 73 6.4 Tight Binding Model . 74 6.4.1 Variational Method . 77 6.5 Time Dependent Perturbation Theory . 78 7 Quantum Mechanics in Crystals 81 7.1 Introduction . 81 7.2 Bloch-Floquet Waves . 82 7.2.1 Periodicity of E(k)............................. 85 7.2.2 Symmetry of E(k) with respect to k ................... 85 Contents iii 7.3 Bloch-Floquet Theorem for 3D . 86 7.4 Fermi-Dirac Distribution Function . 89 7.4.1 Semiconductor, Metal, and Insulator . 90 7.4.2 Why Do Electrons and Holes Conduct Electricity? . 91 7.5 Effective Mass Schr¨odingerEquation . 92 7.6 Heterojunctions and Quantum Wells . 93 7.7 Density of States (DOS) . 94 7.7.1 Fermi Level and Fermi Energy . 96 7.7.2 DOS in a Quantum Well . 96 7.7.3 Quantum Wires . 99 8 Angular Momentum 103 8.1 Introduction . 103 8.1.1 Electron Trapped in a Pill Box . 104 8.1.2 Electron Trapped in a Spherical Box . 106 8.2 Mathematics of Angular Momentum . 109 8.2.1 Transforming to Spherical Coordinates . 110 9 Spin 115 9.1 Introduction . 115 9.2 Spin Operators . 115 9.3 The Bloch Sphere . 118 9.4 Spinor . 118 9.5 Pauli Equation . 119 9.5.1 Splitting of Degenerate Energy Level . 121 9.6 Spintronics . 121 10 Identical Particles 127 10.1 Introduction . 127 10.2 Pauli Exclusion Principle . 128 10.3 Exchange Energy . 129 10.4 Extension to More Than Two Particles . 130 10.5 Counting the Number of Basis States . 132 10.6 Examples . 133 10.7 Thermal Distribution Functions . 134 11 Density Matrix 137 11.1 Pure and Mixed States . 137 11.2 Density Operator . 138 11.3 Time Evolution of the Matrix Element of an Operator . 141 11.4 Two-Level Quantum Systems . 142 11.4.1 Interaction of Light with Two-Level Systems . 143 iv Quantum Mechanics Made Simple 12 Quantization of Classical Fields 153 12.1 Introduction . 153 12.2 The Quantum Harmonic Oscillator Revisited . 154 12.2.1 Eigenfunction by the Ladder Approach . 156 12.3 Quantization of Waves on a Linear Atomic Chain{Phonons . 157 12.4 Schr¨odingerPicture versus Heisenberg Picture . 163 12.5 The Continuum Limit . 164 12.6 Quantization of Electromagnetic Field . 166 12.6.1 Hamiltonian . 168 12.6.2 Field Operators . 169 12.6.3 Multimode Case and Fock State . 171 12.6.4 One-Photon State . 171 12.6.5 Coherent State Revisited . 173 12.7 Thermal Light and Black-Body Radiation . 177 13 Schr¨odingerWave Fields 183 13.1 Introduction . 183 13.2 Fock Space for Fermions . 183 13.3 Field Operators . 185 13.4 Similarity Transform . 187 13.5 Additive One-Particle Operator . ..