QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer

Transcription

1 Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer

2 Contents 1. Historical and Experimental Foundations Introduction and Overview Historically Fundamental Experiments and Insights Particle Properties of Electromagnetic Waves Wave Properties of Particles, Diffraction of Matter Waves Discrete States 8 2. The Wave Function and the Schrodinger Equation The Wave Function and Its Probability Interpretation The Schrodinger Equation for Free Particles Superposition of Plane Waves The Probability Distribution for a Measurement of Momentum Illustration of the Uncertainty Principle Momentum in Coordinate Space Operators and the Scalar Product The Correspondence Principle and the Schrodinger Equation The Correspondence Principle The Postulates of Quantum Theory Many-Particle Systems The Ehrenfest Theorem The Continuity Equation for the Probability Density Stationary Solutions of the Schrodinger Equation, Eigenvalue Equations Stationary States Eigenvalue Equations Expansion in Stationary States The Physical Significance of the Eigenvalues of an Operator Some Concepts from Probability Theory Application to Operators with Discrete Eigenvalues 36

3 X Contents Application to Operators with a Continuous Spectrum Axioms of Quantum Theory Additional Points The General Wave Packet Remark on the Normalizability of the Continuum States 43 Problems One-Dimensional Problems The Harmonie Oscillator The Algebraic Method The Hermite Polynomials The Zero-Point Energy Coherent States Potential Steps Continuity of ip(x) and ip'(x) for a Piecewise Continuous Potential The Potential Step The Tunneling Effect, the Potential Barrier The Potential Barrier The Continuous Potential Barrier Example of Application: a-decay The Potential Well Even Symmetry Odd Symmetry Symmetry Properties Parity Conjugation General Discussion of the One-Dimensional Schrödinger Equation The Potential Well, Resonances Analytic Properties of the Transmission Coefficient The Motion of a Wave Packet Near a Resonance.. 85 Problems The Uncertainty Relation The Heisenberg Uncertainty Relation The Schwarz Inequality The General Uncertainty Relation Energy-Time Uncertainty Passage Time and Energy Uncertainty 98

4 Contents XI Duration of an Energy Measurement and Energy Uncertainty Lifetime and Energy Uncertainty Common Eigenfunctions of Commuting Operators 100 Problems Angular Momentum Commutation Relations, Rotations Eigenvalues of Angular Momentum Operators Orbital Angular Momentum in Polar Coordinates 110 Problems The Central Potential I Spherical Coordinates Bound States in Three Dimensions The Coulomb Potential The Two-Body Problem 135 Problems Motion in an Electromagnetic Field The Hamiltonian Constant Magnetic Field B The Normal Zeeman Effect Canonical and Kinetic Momentum, Gauge Transformation Canonical and Kinetic Momentum Change of the Wave Function Under a Gauge Transformation The Aharonov-Bohm Effect The Wave Function in a Region Free of Magnetic Fields The Aharonov-Bohm Interference Experiment Flux Quantization in Superconductors Free Electrons in a Magnetic Field 152 Problems Operators, Matrices, State Vectors Matrices, Vectors, and Unitary Transformations State Vectors and Dirac Notation The Axioms of Quantum Mechanics Coordinate Representation Momentum Representation Representation in Terms of a Discrete Basis System Multidimensional Systems and Many-Particle Systems

5 XII Contents 8.5 The Schrödinger and Heisenberg Representations The Schrödinger Representation The Heisenberg Representation The Interaction Picture (or Dirac Representation) The Motion of a Free Electron in a Magnetic Field 175 Problems Spin The Experimental Discovery of the Internal Angular Momentum The "Normal" Zeeman Effect The Stern-Gerlach Experiment Mathematical Formulation for Spin-1/ Properties of the Pauli Matrices States, Spinors Magnetic Moment Spatial Degrees of Freedom and Spin 187 Problems Addition of Angular Momenta Posing the Problem Addition of Spin-1/2 Operators Orbital Angular Momentum and Spin 1/ The General Case 196 Problems Approximation Methods for Stationary States Time Independent Perturbation Theory (Rayleigh-Schrödinger) Nondegenerate Perturbation Theory Perturbation Theory for Degenerate States The Variational Principle The WKB (Wentzel-Kramers-Brillouin) Method Brillouin-Wigner Perturbation Theory 209 Problems Relativistic Corrections Relativistic Kinetic Energy Spin-Orbit Coupling The Darwin Term Further Corrections TheLambShift Hyperfine Structure 220 Problems 223

6 Contents XIII 13. Several-Electron Atoms Identical Particles Bosons and Fermions Noninteracting Particles Helium Without the Electron-Electron Interaction Energy Shift Due to the Repulsive Electron-Electron Interaction The Variational Method The Hartree and Hartree-Fock Approximations (Self-consistent Fields) The Hartree Approximation The Hartree-Fock Approximation The Thomas-Fermi Method Atomic Structure and Hund's Rules 249 Problems The Zeeman Effect and the Stark Effect The Hydrogen Atom in a Magnetic Field Weak Field Strong Field, the Paschen-Back Effect The Zeeman Effect for an Arbitrary Magnetic Field Multielectron Atoms Weak Magnetic Field Strong Magnetic Field, the Paschen-Back Effect The Stark Effect Energy Shift of the Ground State Excited States 265 Problems Molecules Qualitative Considerations The Born-Oppenheimer Approximation The Hydrogen Molecular Ion (H2) The Hydrogen Molecule H Energy Levels of a Two-Atom Molecule: Vibrational and Rotational Levels The van der Waals Force 282 Problems 285

7 XIV Contents 16. Time Dependent Phenomena The Heisenberg Picture for a Time Dependent Hamiltonian The Sudden Approximation Time Dependent Perturbation Theory Perturbative Expansion First-Order Transitions Transitions into a Continuous Spectrum, the Golden Rule Periodic Perturbations Interaction with the Radiation Field The Hamiltonian Quantization of the Radiation Field Spontaneous Emission Electric Dipole (El) Transitions Selection Rules for Electric Dipole (El) Transitions The Lifetime for Electric Dipole Transitions Electric Quadrupole and Magnetic Dipole Transitions Absorption and Induced Emission 307 Problems The Central Potential II The Schrödinger Equation for a Spherically Symmetrie Square Well Spherical Bessel Functions Bound States of the Spherical Potential Well The Limiting Case of a Deep Potential Well Continuum Solutions for the Potential Well Expansion of Plane Waves in Spherical Harmonics 319 Problems Scattering Theory Scattering of a Wave Packet and Stationary States The Wave Packet Formal Solution of the Time Independent Schrödinger Equation Asymptotic Behavior of the Wave Packet The Scattering Cross Section Partial Waves The Optical Theorem The Born Approximation Inelastic Scattering 335

8 Contents XV 18.7 Scattering Phase Shirts Resonance Scattering from a Potential Well Low Energy s-wave Scattering; the Scattering Length Scattering at High Energies Additional Remarks Transformation to the Laboratory Frame The Coulomb Potential 348 Problems Supersymmetric Quantum Theory Generalized Ladder Operators Examples Reflection-Free Potentials The <5-function The Harmonie Oscillator The Coulomb Potential Additional Remarks 360 Problems State and Measurement in Quantum Mechanics The Quantum Mechanical State, Causality, and Determinism The Density Matrix The Density Matrix for Pure and Mixed Ensembles The von Neumann Equation Spin-1/2 Systems The Measurement Process The Stern-Gerlach Experiment The Quasiclassical Solution The Stern-Gerlach Experiment as an Idealized Measurement A General Experiment and Coupling to the Environment Influence of an Observation on the Time Evolution Phase Relations in the Stern-Gerlach Experiment The EPR Argument, Hidden Variables, the Bell Inequality The EPR (Einstein-Podolsky-Rosen) Argument The Bell Inequality 385 Problems 388

9 XVI Contents Appendix 391 A. Mathematical Tools for the Solution of Linear Differential Equations 391 A.l The Fourier Transform 391 A.2 The Delta Function and Distributions 391 A.3 Green's Functions 396 B. Canonical and Kinetic Momentum 397 C. Algebraic Determination of the Orbital Angular Momentum Eigenfunctions 398 D. The Periodic Table and Important Physical Quantities Subject Index 409