Lectures on Quantum Mechanics

Transcription

1 Lectures on Quantum Mechanics Steven Weinberg The University of Texas at Austin CAMBRIDGE UNIVERSITY PRESS

2 Contents PREFACE page xv NOTATION xviii 1 HISTORICAL INTRODUCTION Photons 1 Black-body radiation 3 Rayleigh Jeans formula 3 Planck formula LII Atomic constants El Photoelectric effect CI Compton scattering 1.2 Atomic Spectra 5 Discovery of atomic nuclei El Ritz combination principle 3 Bohr quantization condition III Hydrogen spectrum 3 Atomic numbers 3 Sommerfeld quantization condition Einstein A and B coefficients 1.3 Wave Mechanics 11 De Broglie waves III Davi s son Germer experiment El Schrödinger equation 1.4 Matrix Mechanics 14 Radiative transition rate III Harmonic oscillator El Heisenberg matrix algebra Commutation relations ClEquivalence to wave mechanics 1.5 Probabilistic Interpretation 21 Scattering LII Probability density ClExpectation values EI Classical motion E Born rule for transition probabilities Historical Bibliography 27 Problems 27 vii

3 viii Contents 2 PARTICLE STATES IN A CENTRAL POTENTIAL Schrödinger Equation for a Central Potential 29 Hamiltonian for central potentials 13 Orbital angular momentum operators L3 Spectrum of L2 El Separation of wave function El Boundary conditions 2.2 Spherical Harmonics 36 Spectrum of L3 E As sociated Legendre polynomials E Construction of spherical harmonics 3 Orthonormality LI Parity 2.3 The Hydrogen Atom 39 Radial Schrödinger equation 3 Power series solution 13 Laguerre polynomials Energy levels 13 Selection rules 2.4 The Two-Body Problem 44 Reduced masse Relative and center-of-mass coordinates EI Relative and total momenta 13 Hydrogen and deuterium spectra 2.5 The Harmonie Oscillator 45 Separation of wave function 111 Raising and lowering operators Spectrum 2 Normalized wave functions 3 Radiative transition matrix elements Problems 50 3 GENERAL PRINCIPLES OF QUANTUM MECHANICS States 52 Hilbert space Vector spaces E Norms EI Completeness and independence 13 Orthonormalization 13 Probabilities 111Rays LII Dirac notation 3.2 Continuum States 58 From discrete to continuum states EI Normalization E Delta functions 3 Distributions 3.3 Observables 61 Operators LIII Adjoints Cl Matrix representation E Eigenvalues 3 Completeness of eigenvectors E Schwarz inequality EI Uncertainty principle C1 Dyads LII Projection operators EI Density matrix 13 von Neumann entropy 3.4 Symmetries 69 Unitary operators EI Wigner's theoremill Antiunitary operators EI Continuous symmetries EI Commutators 3.5 Space Translation 73 Momentum operators EI Commutation rules EI Momentum eigenstates 13 Bloch waves 13 Band structure

4 Contents ix 3.6 Time Translation 77 Hamiltonians 2 Time-dependent Schrödinger equation 2 Conservation laws 3 Time reversal 3 Galilean invariance 2 Boost generator 3.7 Interpretations of Quantum Mechanics 81 Copenhagen interpretation 111Two classes of interpretationlilmany-worlds interpretations 111 Examples of measurement 2 Decoherence 13 Calculation of probabilities 2 Abandoning realism El Decoherent histories interpretation Problems 96 4 SPIN ET CETERA Rotations 99 Finite rotations 111 Action on physical states 3 Infinitesimal rotations 0 Commutation relations2 Total angular momentumespin 4.2 Angular Momentum Multiplets 104 Raising and lowering operators 2 Spectrum of J2 and J3 2 Spin matrices 3 Pauli matrices [11 J3-independence 13 Stern Gerlach experiment 4.3 Addition of Angular Momenta 109 Choice of basis 2 Clebsch Gordan coefficients 3 Sum rules 2 Hydrogen states El SU(2) formalism 4.4 The Wigner Eckart Theorem 118 Operator transformation properties 2 Theorem for matrix elements 13 Parallel matrix elements 111Photon emission selection rules 4.5 Bosons and Fermions 121 Symmetrical and antisymmetrical states 13 Connection with spin 13 Hartree approximation 2 Pauli exclusion principle C1 Periodic table for atoms EI Magic numbers for nuclei 3 Temperature and chemical potential 2 Statistics 13 Insulators, conductors, semi-conductors 4.6 Internal Symmetries 131 Charge symmetry 111 Isotopic spin symmetry 3 Pions 13 As 3 Strangeness 3 U(1) symmetries 2 SU (3) symmetry 4.7 Inversions 138 Space Inversion 13 Orbital parity 3 Intrinsic parity 3 Parity of pions 2 Violations of parity conservation 2 P, C, and T 4.8 Algebraic Derivation of the Hydrogen Spectrum 142 Runge Lenz vector 0 SO(3) 0 SO(3) commutation relations 3 Energy levels 2 Scattering states Problems 146

5 Contents 5 APPROXIMATIONS FOR ENERGY EIGENVALUES First-Order Perturbation Theory 148 Energy shift E Dealing with degeneracy E State vector perturbation E A classical analog 5.2 The Zeeman Effect 152 Gyromagnetic ratio E Lande g-factor El Sodium D lines E Normal and anomalous Zeeman effect E Paschen Back effect 5.3 The First-Order Stark Effect 157 Mixing of 2s1/2 and 2p1/2 states E Energy shift for weak fields 111 Energy shift for strong fields 5.4 Second-Order Perturbation Theory 160 Energy shift El Ultraviolet and infrared divergences El Closure approximation Second-order Stark effect 5.5 The Variational Method 162 Upper bound on ground state energy ClApproximation to state vectors E Virial theorem El Other states 5.6 The Born Oppenheimer Approximation 165 Reduced Hamiltonian E Hellmann Feynman theorem E Estimate of corrections E Electronic, vibrational, and rotational modes El Effective theories 5.7 The WKB Approximation 171 Approximate solutions E Validity conditions E Turning points El Energy eigenvalues one dimension E Energy eigenvalues three dimensions 5.8 Broken Symmetry 179 Approximate solutions for thick barriers E Energy splitting El Decoherence El Oscillations LIChiral molecules Problems APPROXIMATIONS FOR TIME-DEPENDENT PROBLEMS First-Order Perturbation Theory 183 Differential equation for amplitudes E Approximate solution 6.2 Monochromatic Perturbations 184 Transition rate III Fermi golden rulell1continuum final states 6.3 Ionization by an Electromagnetic Wave 187 Nature of perturbation 111Conditions on frequency El Ionization rate of hydrogen ground state

6 Contents xi 6.4 Fluctuating Perturbations 189 Stationary fluctuations Correlation function 173 Transition rate 6.5 Absorption and Stimulated Emission of Radiation 191 Dipole approximation fl Transition rates Energy density of radiation EI B-coefficients 111Spontaneous transition rate 6.6 The Adiabatic Approximation 193 Slowly varying HamiltonianslilDynamical phase 3 Non-dynamical phaselildegenerate case 6.7 The Berry Phase 196 Geometric character of the non-dynamical phase EI Closed curves in parameter space 13 General formula for the Berry phase [3 Spin in a slowly varying magnetic field Problems POTENTIAL SCATTERING In-States 203 Wave packets LII Lippmann Schwinger equation 111 Wave packets at early times 111 Spread of wave packet 7.2 Scattering Amplitudes 208 Green's function for scattering 3 Definition of scattering amplitude EI Wave packet at late times EI Differential cross-section 7.3 The Optical Theorem 211 Derivation of theorem111conservation of probabilitylildiffraction peak 7.4 The Born Approximation 214 First-order scattering amplitude 3 Scattering by shielded Coulomb potential 7.5 Phase Shifts 216 Partial wave expansion of plane wave EI Partial wave expansion of "in" wave functione Partial wave expansion of scattering amplitude El Scattering cross-section EI Scattering length and effective range 7.6 Resonances 220 Thick barriers 111 Breit Wigner formula 111 Decay rate EI Alpha decay 111 Ramsauer Townsend effect 7.7 Time Delay 224 Wigner formulallicausality 7.8 Levinson's Theorem 226 Conservation of discrete states111growth of phase shift

7 xii Contents 7.9 Coulomb Scattering 227 Separation of wave function 3 Kummer functions E Scattering amplitude 7.10 The Eikonal Approximation 229 WKB approximation in three dimensions 3 Initial surface E Ray paths E Calculation of phase D Calculation of amplitude E Application to potential scattering Problems GENERAL SCATTERING THEORY The S-Matrix 235 "In" and "out " states 13 Wave packets at early and late times Ei Definition of the S-Matrix E Normalization of the "in" and "out" states Ei Unitarity of the S-matrix 8.2 Rates 240 Transition probabilities in a spacetime box E Decay rates ü Cross-sections E Relative velocity 13 Connection with scattering amplitudes 13 Final states 8.3 The General Optical Theorem 244 Optical theorem for multi -particle states E Two-particle case 8.4 The Partial Wave Expansion 245 Discrete basis for two-particle states Ei Two-particle S-matrix Ei Total and scattering cross-sections 13 Phase shifts Ei High-energy scattering 8.5 Resonances Revisited 252 S-matrix near a resonance energy Ei Consequences of unitaiity E General Breit Wigner formula 0 Total and scattering cross-sections Ei Branching ratios 8.6 Old-Fashioned Perturbation Theory 256 Perturbation series for the S-matrix Ei Functional analysis13 Square-integrable kerne113 Sufficient conditions for convergence Ei Upper bound on binding energies E Distorted wave Born approximation E Coulomb suppression 8.7 Time-Dependent Perturbation Theory 262 Time-development Operator Ei Interaction picture Ei Time-ordered products D Dyson perturbation series D Lorentz invariance Ei "In-in" formalism 8.8 Shallow Bound States 267 Low equation Ei Low-energy approximation Ei Solution for scattering length 13 Neutron proton scattering Ei Solution using Herglotz theorem Problems 273

8 Contents xiii 9 THE CANONICAL FORMALISM The Lagrangian Formalism 276 Stationary action El Lagrangian equations of motion 3 Example: spherical coordinates 9.2 Symmetry Principles and Conservation Laws 278 Noether's theorem 3 Conserved quantities from symmetries of Lagrangian 3 Space translation 3 Rotations El Symmetries of action 9.3 The Hamiltonian Formalism 279 Time translation and Hamiltonian D Hamiltonian equations of motion 3 Spherical coordinates again 9.4 Canonical Commutation Relations 281 Conserved quantities as symmetry generators 3 Commutators of canonical variables and conjugates El Momentum and angular momentum El Poisson brackets 3 Jacobi identity 9.5 Constrained Hamiltonian Systems 285 Example: particle on a surface El Primary and secondary constraints EI First- and second-class constraints El Dirac brackets 9.6 The Path-Integral Formalism 290 Derivation of the general path integral El Integrating out momenta El The free particle 3 Two-slit experiment El Interactions Problems CHARGED PARTICLES IN ELECTROMAGNETIC FIELDS Canonical Formalism for Charged Particles 298 Equations of motion El Scalar and vector potentials 3 Lagrangian D Hamiltonian El Commutation relations 10.2 Gauge Invariance 300 Gauge transformations of potentials EI Gauge transformation of Lagrangian 3 Gauge transformation of Hamiltonian El Gauge transformation of state vector 3 Gauge invariance of energy eigenvalues 10.3 Landau Energy Levels 302 Hamiltonian in a uniform magnetic field El Energy levels 3 Near degeneracy El Fermi level El Periodicity in 1/ B, III Shubnikow de Haas and de Haas van Alphen effects 10.4 The Aharonov Bohm Effect 305 Application of the eikonal approximation D Interference between alternate ray paths III Relation to Berry phase III Effect of field-free vector potential 3 Periodicity in the flux Problems 307

9 xiv Contents 11 THE QUANTUM THEORY OF RADIATION The Euler Lagrange Equations 309 General field theories3 Variational derivatives of Lagrangian El Lagrangian density 11.2 The Lagrangian for Electrodynamics 311 Maxwell equations 3 Charge density and current density El Field, interaction, and matter Lagrangians 11.3 Commutation Relations for Electrodynamics 313 Coulomb gauge El ConstraintsClApplying Dirac brackets 11.4 The Hamiltonian for Electrodynamics 316 Evaluation of Hamiltonian3 Coulomb energy El Recovery of Maxwell's equations 11.5 Interaction Picture 318 Interaction picture operators 3 Expansion in plane wavescipolarization vectors E 11.6 Photons 322 Creation and annihilation operators E Fock space El Photon energies 3 Vacuum energy 17) Photon momentum El Photon spin El Varieties of polarization El Coherent states 11.7 Radiative Transition Rates 327 S-matrix for photon emission CISeparation of center-of-mass motion E General decay rate III Electric dipole radiationcielectric quadrupole and magnetic dipole radiation 21 cm radiation E No 0 > 0 transitions Problems ENTANGLEMENT Paradoxes of Entanglement 336 The Einstein Podolsky Rosen paradox 3 The Bohm paradox El Instantaneous communication? El Entanglement entropy 12.2 The Bell Inequalities 341 Local hidden variable theories CI Two-spin inequality El Generalized inequality Experimental tests 12.3 Quantum Computation 346 Qbits CI Comparison with classical digital computers El Computation as unitary transformation El Fourier transforms III Gates El Reading the memory E No-copying theorem El Necessity of entanglement AUTHOR INDEX 350 SUBJECT INDEX 353