Principles Of Quantum Mechanics

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Preface to the Second Edition Preface to the First Edition For Whom Is this Book Intended? Acknowledgments Prelude Note to the Student Contents 1 Mathematical Introduction 1.1. Linear Vector Spaces: Basics 1.2. Inner Product Spaces 1.3. Dual Spaces and the Dirac Notation 1.3.1. Expansion of Vectors in an Orthonormal Basis 1.3.2. Adjoint Operation Gram-Schmidt Theorem Schwarz and Triangle Inequalities 1.4. Subspaces 1.5. Linear Operators 1.6. Matrix Elements of Linear Operators Matrices Corresponding to Products of Operators The Adjoint of an Operator Hermitian, Anti-Hermitian, and Unitary Operators 1. 7. Active and Passive Transformations 1.8. The Eigenvalue Problem The Characteristic Equation and the Solution to the Eigenvalue Problem Degeneracy Diagonalization of Hermitian Matrices Simultaneous Diagonalization of Two Hermitian Operators The Normal Modes 1.9. Functions of Operators and Related Concepts Derivatives of Operators with Respect to Parameters 1.10. Generalization to Infinite Dimensions Operators in Infinite Dimensions 2 Review of Classical Mechanics 2.1. ·The Principle of Least Action and Lagrangian Mechanics 2.2. The Electromagnetic Lagrangian 2.3. The Two-Body Problem 2.4. How Smart Is a Particle? 2.5. The Hamiltonian Formalism 2.6. The Electromagnetic Force in the Hamiltonian Scheme 2.7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations Canonical Transformations Active Transformations 2.8. Symmetries and Their Consequences A Useful Relation Between SandE 3 All Is Not Well with Classical Mechanics 3.1. Particles and Waves in Classical Physics 3.2. An Experiment with Waves and Particles (Classical) 3.3. The Double-Slit Experiment with Light 3.4. Matter Waves (de Broglie Waves) 3.5. Conclusions 4 The Postulates-a General Discussion 4.1. The Postulatest 4.2. Discussion of Postulates I-III CoUapse of the State Vector How to Test Quantum Theory Expectation Value The Uncertainty Compatible and Incompatible Variables The Density Matrix-a Digressiont Generalization to More Degrees of Freedom 4.3. The Schrodinger Equation (Dotting Your i 's and Crossing Your h's) Setting Up the Schrödinger Equation General Approach to the Solution Choosing a Basis for Solving Schrodinger's Equation 5 Simple Problems in One Dimension 5.1. The Free Particle Time Evolution of the Gaussian Packet Some General Features of Energy Eigenfunctions 5.2. The Particle in a Box 5.3. The Continuity Equation for Probability Ensemble Interpretation of j 5.4. The Single-Step Potential: A Problem in Scattering 5.5. The Double-Slit Experiment 5.6. Some Theorems 6 The Classical Limit 7 The Harmonic Oscillator 7.1. Why Study the Harmonic Oscillator 7 .2. Review of the Classical OscUla tor 7.3. Quantization of the Oscillator (Coordinate Basis) 7 .4. The OsciUator in the Energy Basis 7 .5. Passage from the Energy Basis to the X Basis 8 The Path Integral Formulation of Quantum Theory 8.1. The Path Integral Recipe 8.2. Analysis of the Recipe 8.3. An Approximation to U(t) for a Free Particle 8.4. Path Integral Evaluation of the Free-Particle Propagator 8.5. Equivalence to the Schrodinger Equation 8.6. Potentials of the Form V =a + bx + cx2 + di +exx 9 The Heisenberg Uncertainty Relations 9.1. Introduction 9.2. Derivation of the Uncertainty Relations 9.3. The Minimum Uncertainty Packet 9.4. Applications of the Uncertainty Principle 9.5. The Energy-Time Uncertainty Relation 10 Systems with N Degrees of Freedom 10.1. N Particles in One Dimension The Two-Particle Hilbert Space V 102 As a Direct Product Space The Direct Product Revisited Evolution of the Two-Particle State Vector N Particles in One Dimension 10.2. More Particles in More Dimensions 10.3. Identical Particles The Classical Case Two-Particle Systems-Symmetric and Antisymmetric States Bosons and Fermions Bosonic and Fermionic Hilbert Spaces Determination of Particle Statistics Systems of N Identical Particles When Can We Ignore Symmetrization and Antisymmetrization? 11 Symmetries and Their Consequences 11.1. Overview 11.2. Translational Invariance in Quantum Theory Translation in Terms of Passive Transformations A Digression on the Analogy with Classical Mechanicst Finite Translations A Digression on Finite Canonical and Unitary Transformations System of Particles Implications of Translational Invariances 11.3. Time Translational Invariance 11.4. Parity Invariance 11.5. Time-Reversal Symmetry 12 Rotational Invariance and Angular Momentum 12.1. Translations in Two Dimensions 12.2. Rotations in Two Dimensions Explicit Construction of U[R] 12.3. The Eigenvalue Problem of Lz Solutions to Rotationally Invariant Problems 12.4. Angular Momentum in Three Dimensions 12.5. The Eigenvalue Problem of L 2 and Lz Finite Rotations Angular Momentum Eigenfunctions in the Coordinate Basis 12.6. Solution of Rotationally Invariant Problems General Properties of U Et The Free Particle in Spherical Coordinates Connection with the Solution in Cartesian Coordinates The Isotropic Oscillator 13 The Hydrogen Atom 13.1. The Eigenvalue Problem The Energy Levels The Wave Functions 13.2. The Degeneracy of the Hydrogen Spectrum 13.3. Numerical Estimates and Comparison with Experiment Numerical Estimates Comparison with Experiment 13.4. Multielectron Atoms and the Periodic Table 14 Spin 14.1. Introduction 14.2. What is the Nature of Spin? 14.3. Kinematics of Spin Explicit Forms of Rotation Operators 14.4. Spin Dynamics Orbital Magnetic Moment in Quantum Theory Spin Magnetic Moment Paramagnetic Resonance Negative Absolute Temperature (Optional Digression) 14.5. Return of Orbital Degrees of Freedom The Stern-Gerlach (SG) Experiment 15 Addition of Angular Momenta 15.1. A Simple Example 15.2. The General Problem Clebsch-Gordan (CG) Coefficients Addition of L and S The Modified Spectroscopic Notation 15.3. Irreducible Tensor Operators Tensor Operators 15.4. Explanation of Some "Accidental" Degeneracies Hydrogen The Oscillator The Free-Particle Solutions 16 The Variational and WKB Methods 16.1. The Variational Method 16.2. The Wentzel-Kramers--Brillouin Method Connection with the Path Integral Formalism Tunneling Amplitudes Bound States 17 Time-Independent Perturbation Theory 17.1. The Formalism 17.2. Some Examples Selection Rules 17.3. Degenerate Perturbation Theory Fine Structure 18 Time-Dependent Perturbation Theory 18.1. The Problem 18.2. First-Order Perturbation Theory The Sudden Perturbation The Adiabatic Perturbation The Periodic Perturbation 18.3. Higher Orders in Perturbation Theory The Interaction Picture The Heisenberg Picture 18.4. A General Discussion of Electromagnetic Interactions Classical Electrodynamics The Potentials in Quantum Theory 18.5. Interaction of Atoms with Electromagnetic Radiation Photoelectric Effect in Hydrogen Field Quantization Spontaneous Decay 19 Scattering Theory 19.1. Introduction 19.2. Recapitulation of One-Dimensional Scattering and Overview 19.3. The Born Approximation (Time-Dependent Description) 19.4. Born Again (The Time-Independent Description) Validity of the Born Approximation 19.5. The Partial Wave Expansion A Model Calculation of δ1: The Hard Sphere Resonances 19.6. Two-Particle Scattering Passage to the Lab Frame Scattering of Identical Particles 20 The Dirac Equation 20.1. The Free-Particle Dirac Equation 20.2. Electromagnetic Interaction of the Dirac Particle The Electron Spin and Magnetic Moment Hydrogen Fine Structure 20.3. More on Relativistic Quantum Mechanics 21 Path Integrals: Part II 21.1. Derivation of the Path Integral The Landau Levels The Berry Phase Coherent State Path Integral 21.2. Imaginary Time Formalism Path Integral for the Imaginary Time Propagator Tunneling by Path Integrals: Well, well! Spontaneous Symmetry Breaking Imaginary Time Path Integrals and Quantum Statistical Mechanics Relation to Classical Statistical Mechanics 21.3. Spin and Fermion Path Integrals Spin Coherent States and Path Integral Fermion Oscillator and Coherent States The Fermionic Path Integral 21.4. Summary Bibliography Appendix A.1. Matrix Inversion A.2. Gaussian Integrals A.3. Complex Numbers A.4. The iε Prescription Answers to Selected Exercises Table of Constants Index