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Supplementary Lecture Notes: Quantum Mechanics Made Simple

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Supplementary Lecture Notes: Quantum Mechanics Made Simple Quantum Mechanics Made Simple: Lecture Notes Weng Cho CHEW1 October 5, 2012 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 7 2.1 Introduction . .7 2.2 Lagrangian Formulation . .8 2.3 Hamiltonian Formulation . 10 2.4 More on Hamiltonian . 12 2.5 Poisson Bracket . 12 3 Quantum Mechanics|Some Preliminaries 15 3.1 Introduction . 15 3.2 Probabilistic Interpretation of the wavefunction . 16 3.3 Simple Examples of Time Independent Schr¨odingerEquation . 19 3.3.1 Particle in a 1D Box . 19 3.3.2 Particle Scattering by a Barrier . 21 3.3.3 Particle in a Potential Well . 21 3.4 The Quantum Harmonic Oscillator . 23 4 Time-Dependent Schr¨odingerEquation 27 4.1 Introduction . 27 4.2 Quantum States in the Time Domain . 27 4.3 Coherent State . 28 4.4 Measurement Hypothesis and Expectation Value . 29 4.4.1 Time Evolution of the Hamiltonian Operator . 31 4.4.2 Uncertainty Principle . 32 4.4.3 Particle Current . 32 i ii Quantum Mechanics Made Simple 5 Mathematical Preliminaries 35 5.1 A Function is a Vector . 35 5.2 Operators . 38 5.2.1 Matrix Representation of an Operator . 38 5.2.2 Bilinear Expansion of an Operator . 39 5.2.3 Trace of an Operator . 39 5.2.4 Unitary Operators . 41 5.2.5 Hermitian Operators . 42 5.3 Identity Operator in a Continuum Space . 44 5.4 Changing Between Representations . 47 5.4.1 The Coordinate Basis Function . 48 5.5 Commutation of Operators . 49 5.6 Expectation Value and Eigenvalue of Operators . 49 5.7 Generalized Uncertainty Principle . 51 5.8 Time Evolution of the Expectation Value of an Operator . 53 5.9 Periodic Boundary Condition . 54 6 Approximate Methods in Quantum Mechanics 57 6.1 Introduction . 57 6.2 Use of an Approximate Subspace . 57 6.3 Time Independent Perturbation Theory . 59 6.4 Tight Binding Model . 64 6.4.1 Variational Method . 66 6.4.2 Time Dependent Perturbation Theory . 67 7 Quantum Mechanics in Crystals 71 7.1 Introduction . 71 7.2 Bloch-Floquet Waves . 72 7.3 Bloch-Floquet Theorem for 3D . 74 7.4 Effective Mass Schr¨odingerEquation . 78 7.5 Density of States (DOS) . 79 7.6 DOS in a Quantum Well . 81 8 Angular Momentum 85 8.1 Introduction . 85 8.1.1 Electron Trapped in a Pill Box . 86 8.1.2 Electron Trapped in a Spherical Box . 88 8.2 Mathematics of Angular Momentum . 91 8.3 L^2 Operator . 92 Contents iii 9 Spin 95 9.1 Introduction . 95 9.2 Spin Operators . 95 9.3 The Bloch Sphere . 97 9.4 Spinor . 98 9.5 Pauli Equation . 99 10 Identical Particles 101 10.1 Introduction . 101 10.2 Pauli Exclusion Principle . 102 10.3 Exchange Energy . 102 10.4 Extension to More Than Two Particles . 104 10.5 Counting the Number of Basis states . 105 10.6 Examples . 106 10.7 Thermal Distribution Functions . 107 11 Density Matrix 111 11.1 Pure and Mixed States . 111 11.2 Density Operator . 112 11.3 Time Evolution of the Matrix Element of an Operator . 114 11.4 Interaction of Light with Two-Level Atomic System . 115 12 Quantization of Classical Fields 123 12.1 Introduction . 123 12.2 The Quantum Harmonic Oscillator Revisited . 124 12.2.1 Eigenfunction by the Ladder Approach . 126 12.3 Quantization of Waves on a Linear Atomic Chain{Phonons . 127 12.4 Schr¨odingerPicture versus Heisenberg Picture . 132 12.5 The Continuum Limit . 133 12.6 Quantization of Electromagnetic Field . 136 12.6.1 Hamiltonian . 137 12.6.2 Field Operators . 138 12.6.3 Multimode Case and Fock State . 139 12.6.4 One-Photon State . 140 12.6.5 Coherent State Revisited . 141 13 Schr¨odingerWave Fields 145 13.1 Introduction . 145 13.2 Fock Space for Fermions . 145 13.3 Field Operators . 147 13.4 Similarity Transform . 149 13.5 Additive One-Particle Operator . 150 13.5.1 Three-Particle Case . 151 13.6 Additive Two-Particle Operator . 153 13.7 More on Field Operators . 155 iv Quantum Mechanics Made Simple 13.8 Boson Wave Field . 157 13.9 Boson Field Operators . 158 13.10Additive One-Particle Operator . 159 13.11The Difference between Boson Field and Photon Field . 160 14 Interaction of Different Particles 161 14.1 Introduction . 161 14.2 Interaction of Particles . 161 14.3 Time-Dependent Perturbation Theory . 162 14.3.1 Absorption . 163 14.4 Spontaneous Emission . 165 14.5 Stimulated Emission . 165 14.6 Multi-photon Case . 167 14.7 Total Spontaneous Emission Rate . 167 15 Quantum Information and Quantum Interpretation 169 15.1 Introduction . 169 15.2 Quantum Cryptography . 169 15.2.1 No-cloning Theorem . 169 15.2.2 Entangled States . 170 15.2.3 A Simple.
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