QUANTUM MECHANICS

By ELIZABETH LEE

INTRODUCTION AND OVERVIEW HISTORICALLY FUNDAMENTAL EXPERIMENTS AND INSIGHTS PARTICLE PROPERTIES OF ELECTROMAGNETIC WAVES WAVE PROPERTIES OF PARTICLES, DI?RACTION OF MATTER WAVES DISCRETE STATES 2 THE WAVE FUNCTION AND THE SCHR¨ODINGER EQUATION THE WAVE FUNCTION AND ITS PROBABILITY INTERPRETATION THE SCHR¨ODINGER 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 SCHR¨ODINGER 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 SCHR¨ODINGER EQUATION, EIGENVALUE EQUATIONS STATIONARY STATES EIGENVALUE EQUATIONS EXPANSION IN STATIONARY STATES THE PHYSICAL SIGNI?CANCE OF THE EIGENVALUES OF AN OPERATOR SOME CONCEPTS FROM PROBABILITY THEORY APPLICATION TO OPERATORS WITH DISCRETE EIGENVALUES 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 PROBLEMS 3 ONE-DIMENSIONAL PROBLEMS THE HARMONIC OSCILLATOR THE ALGEBRAIC METHOD THE HERMITE POLYNOMIALS THE ZERO-POINT ENERGY COHERENT STATES POTENTIAL STEPS CONTINUITY OF ?(X) AND ??(X) FOR A PIECEWISE CONTINUOUS POTENTIAL THE POTENTIAL STEP THE TUNNELING E?ECT, THE POTENTIAL BARRIER THE POTENTIAL BARRIER THE CONTINUOUS POTENTIAL BARRIER EXAMPLE OF APPLICATION: ?-DECAY THE POTENTIAL WELL EVEN SYMMETRY ODD SYMMETRY SYMMETRY PROPERTIES PARITY CONJUGATION GENERAL DISCUSSION OF THE ONE-DIMENSIONAL SCHR¨ODINGER EQUATION THE POTENTIAL WELL, RESONANCES ANALYTIC PROPERTIES OF THE TRANSMISSION COE?CIENT THE MOTION OF A WAVE PACKET NEAR A RESONANCE 4 THE UNCERTAINTY RELATION THE HEISENBERG UNCERTAINTY RELATION THE SCHWARZ INEQUALITY THE GENERAL UNCERTAINTY RELATION ENERGY–TIME UNCERTAINTY PASSAGE TIME AND ENERGY UNCERTAINTY DURATION OF AN ENERGY MEASUREMENT AND ENERGY UNCERTAINTY LIFETIME AND ENERGY UNCERTAINTY COMMON EIGENFUNCTIONS OF COMMUTING OPERATORS 5 ANGULAR MOMENTUM COMMUTATION RELATIONS, ROTATIONS EIGENVALUES OF ANGULAR MOMENTUM OPERATORS ORBITAL ANGULAR MOMENTUM IN POLAR COORDINATES 6 THE CENTRAL POTENTIAL I SPHERICAL COORDINATES BOUND STATES IN THREE DIMENSIONS THE COULOMB POTENTIAL THE TWO-BODY PROBLEM 7 MOTION IN AN ELECTROMAGNETIC FIELD THE HAMILTONIAN CONSTANT MAGNETIC FIELD B THE NORMAL ZEEMAN E?ECT CANONICAL AND KINETIC MOMENTUM, GAUGE TRANSFORMATION CANONICAL AND KINETIC MOMENTUM CHANGE OF THE WAVE FUNCTION UNDER A GAUGE TRANSFORMATION THE AHARONOV–BOHM E?ECT 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 8 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 THE SCHR¨ODINGER, HEISENBERG AND INTERACTION REPRESENTATIONS THE SCHR¨ODINGER REPRESENTATION THE HEISENBERG REPRESENTATION THE INTERACTION PICTURE (OR DIRAC REPRESENTATION) THE MOTION OF A FREE ELECTRON IN A MAGNETIC FIELD 9 SPIN THE EXPERIMENTAL DISCOVERY OF THE INTERNAL ANGULAR MOMENTUM THE “NORMAL” ZEEMAN E?ECT THE STERN–GERLACH EXPERIMENT MATHEMATICAL FORMULATION FOR SPIN-1/2 PROPERTIES OF THE PAULI MATRICES STATES, SPINORS MAGNETIC MOMENT SPATIAL DEGREES OF FREEDOM AND SPIN 10 ADDITION OF ANGULAR MOMENTA POSING THE PROBLEM ADDITION OF SPIN-1/2 OPERATORS ORBITAL ANGULAR MOMENTUM AND SPIN 1/2 THE GENERAL CASE 11 APPROXIMATION METHODS FOR STATIONARY STATES TIME INDEPENDENT PERTURBATION THEORY (RAYLEIGH–SCHR¨ODINGER) NONDEGENERATE PERTURBATION THEORY PERTURBATION THEORY FOR DEGENERATE STATES THE VARIATIONAL PRINCIPLE THE WKB (WENTZEL–KRAMERS–BRILLOUIN) METHOD BRILLOUIN–WIGNER PERTURBATION THEORY 12 RELATIVISTIC CORRECTIONS RELATIVISTIC KINETIC ENERGY SPIN–ORBIT COUPLING THE DARWIN TERM FURTHER CORRECTIONS THE LAMB SHIFT HYPER?NE STRUCTURE 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 14 THE ZEEMAN E?ECT AND THE STARK E?ECT THE HYDROGEN ATOM IN A MAGNETIC FIELD WEAK FIELD STRONG FIELD, THE PASCHEN–BACK E?ECT THE ZEEMAN E?ECT FOR AN ARBITRARY MAGNETIC FIELD MULTIELECTRON ATOMS WEAK MAGNETIC FIELD STRONG MAGNETIC FIELD, THE PASCHEN–BACK E?ECT THE STARK E?ECT ENERGY SHIFT OF THE GROUND STATE EXCITED STATES 15 MOLECULES QUALITATIVE CONSIDERATIONS THE BORN–OPPENHEIMER APPROXIMATION THE HYDROGEN MOLECULAR ION (H + 2 ) THE HYDROGEN MOLECULE H2 ENERGY LEVELS OF A TWO-ATOM MOLECULE: VIBRATIONAL AND ROTATIONAL LEVELS THE VAN DER WAALS FORCE 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 QUANTIZATION OF THE RADIATION FIELD SPONTANEOUS EMISSION ELECTRIC DIPOLE (E1) TRANSITIONS SELECTION RULES FOR ELECTRIC DIPOLE (E1) TRANSITIONS THE LIFETIME FOR ELECTRIC DIPOLE TRANSITIONS ELECTRIC QUADRUPOLE AND MAGNETIC DIPOLE TRANSITIONS ABSORPTION AND INDUCED EMISSION 17 THE CENTRAL POTENTIAL II THE SCHR¨ODINGER EQUATION FOR A SPHERICALLY SYMMETRIC 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 18 SCATTERING THEORY SCATTERING OF A WAVE PACKET AND STATIONARY STATES THE WAVE PACKET FORMAL SOLUTION OF THE TIME INDEPENDENT SCHR¨ODINGER EQUATION ASYMPTOTIC BEHAVIOR OF THE WAVE PACKET THE SCATTERING CROSS SECTION PARTIAL WAVES THE OPTICAL THEOREM THE BORN APPROXIMATION INELASTIC SCATTERING SCATTERING PHASE SHIFTS RESONANCE SCATTERING FROM A POTENTIAL WELL LOW ENERGY S-WAVE SCATTERING; THE SCATTERING LENGTH 18 10 SCATTERING AT HIGH ENERGIES 18 11 ADDITIONAL REMARKS 18 11 1 TRANSFORMATION TO THE LABORATORY FRAME 18 11 2 THE COULOMB POTENTIAL 19 SUPERSYMMETRIC QUANTUM THEORY GENERALIZED LADDER OPERATORS EXAMPLES RE?ECTION-FREE POTENTIALS THE ?-FUNCTION ADDITIONAL REMARKS 20 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 QUASICLASSICAL SOLUTION THE STERN–GERLACH EXPERIMENT AS AN IDEALIZED MEASUREMENT A GENERAL EXPERIMENT AND COUPLING TO THE ENVIRONMENT IN?UENCE 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