BASIC QUANTUM MECHANICS
CONTENTS BASIC FORMALISM STATE VECTORS OPERATORS AND PHYSICAL OBSERVABLES EIGENSTATES HERMITIAN CONJUGATION AND HERMITIAN OPERATORS HERMITIAN OPERATORS: THEIR EIGENSTATES AND EIGENVALUES SUPERPOSITION PRINCIPLE COMPLETENESS RELATION UNITARY OPERATORS UNITARY OPERATORS AS TRANSFORMATION OPERATORS MATRIX FORMALISM EIGENSTATES AND DIAGONALIZATION OF MATRICES DENSITY OPERATOR MEASUREMENT PROBLEMS FUNDAMENTAL COMMUTATOR AND TIME EVOLUTION OF STATE VECTORS AND OPERATORS CONTINUOUS VARIABLES: X AND P OPERATORS CANONICAL COMMUTATOR [X , P] P AS A DERIVATIVE OPERATOR: ANOTHER WAY X AND P AS HERMITIAN OPERATORS UNCERTAINTY PRINCIPLE SOME INTERESTING APPLICATIONS OF UNCERTAINTY RELATIONS SPACE DISPLACEMENT OPERATOR TIME EVOLUTION OPERATOR APPENDIX TO CHAPTER DYNAMICAL EQUATIONS SCHRÖDINGER PICTURE HEISENBERG PICTURE INTERACTION PICTURE SUPERPOSITION OF TIME-DEPENDENT STATES AND ENERGY–TIME UNCERTAINTY RELATION TIME DEPENDENCE OF THE DENSITY OPERATOR PROBABILITY CONSERVATION EHRENFEST’S THEOREM FREE PARTICLES FREE PARTICLE IN ONE DIMENSION NORMALIZATION MOMENTUM EIGENFUNCTIONS AND FOURIER TRANSFORMS MINIMUM UNCERTAINTY WAVE PACKET GROUP VELOCITY OF A SUPERPOSITION OF PLANE WAVES THREE DIMENSIONS – CARTESIAN COORDINATES THREE DIMENSIONS – SPHERICAL COORDINATES THE RADIAL WAVE EQUATION PROPERTIES OF YLM(?, ?) ANGULAR MOMENTUM DETERMINING L2 FROM THE ANGULAR VARIABLES COMMUTATOR LI, LJ AND L2, LJ LADDER OPERATORS PARTICLES WITH SPIN ½ SPIN ½ SYSTEM PAULI MATRICES THE SPIN ½ EIGENSTATES MATRIX REPRESENTATION OF ? X AND ? Y EIGENSTATES OF ? X AND ? Y EIGENSTATES OF SPIN IN AN ARBITRARY DIRECTION SOME IMPORTANT RELATIONS FOR ? I ARBITRARY 2 × 2 MATRICES IN TERMS OF PAULI MATRICES PROJECTION OPERATOR FOR SPIN ½ SYSTEMS DENSITY MATRIX FOR SPIN ½ STATES AND THE ENSEMBLE AVERAGE COMPLETE WAVEFUNCTION PAULI EXCLUSION PRINCIPLE AND FERMI ENERGY GAUGE INVARIANCE, ANGULAR MOMENTUM, AND SPIN GAUGE INVARIANCE QUANTUM MECHANICS CANONICAL AND KINEMATIC MOMENTA INTERACTION WITH THE ORBITAL ANGULAR MOMENTUM INTERACTION WITH SPIN: INTRINSIC MAGNETIC MOMENT SPIN–ORBIT INTERACTION AHARONOV–BOHM EFFECT STERN–GERLACH EXPERIMENTS EXPERIMENTAL SET-UP AND ELECTRON’S MAGNETIC MOMENT DISCUSSION OF THE RESULTS SOME EXACTLY SOLVABLE BOUND-STATE PROBLEMS SIMPLE ONE-DIMENSIONAL SYSTEMS DELTA-FUNCTION POTENTIAL PROPERTIES OF A SYMMETRIC POTENTIAL THE AMMONIA MOLECULE PERIODIC POTENTIALS PROBLEMS IN THREE DIMENSIONS SIMPLE SYSTEMS HYDROGEN-LIKE ATOM HARMONIC OSCILLATOR HARMONIC OSCILLATOR IN ONE DIMENSION COHERENT STATES EIGENSTATES OF THE LOWERING OPERATOR COHERENT STATES AND SEMICLASSICAL DESCRIPTION INTERACTION OF A HARMONIC OSCILLATOR WITH AN ELECTRIC FIELD TWO-DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR THE TWO-DIMENSIONAL HAMILTONIAN LANDAU LEVELS AND QUANTUM HALL EFFECT LANDAU LEVELS IN SYMMETRIC GAUGE WAVEFUNCTIONS FOR THE LLL LANDAU LEVELS IN LANDAU GAUGE QUANTUM HALL EFFECT WAVEFUNCTION FOR FILLED LLLS IN A FERMI SYSTEM TWO-LEVEL PROBLEMS TIME-INDEPENDENT PROBLEMS TIME-DEPENDENT PROBLEMS SPIN ½ SYSTEMS IN THE PRESENCE OF MAGNETIC ?ELDS CONSTANT MAGNETIC FIELD SPIN PRECESSION TIME-DEPENDENT MAGNETIC FIELD: SPIN MAGNETIC RESONANCE OSCILLATION AND REGENERATION IN NEUTRINOS AND NEUTRAL K-MESONS AS TWO-LEVEL SYSTEMS NEUTRINOS THE SOLAR NEUTRINO PUZZLE NEUTRINO OSCILLATIONS DECAY AND REGENERATION OSCILLATION AND REGENERATION OF STABLE AND UNSTABLE SYSTEMS NEUTRAL K-MESONS TIME-INDEPENDENT PERTURBATION FOR BOUND STATES HARMONIC OSCILLATOR: PERTURBATIVE VS EXACT RESULTS SECOND-ORDER STARK EFFECT DEGENERATE STATES LINEAR STARK EFFECT TIME-DEPENDENT PERTURBATION HARMONIC PERTURBATION AND FERMI’S GOLDEN RULE TRANSITIONS INTO A GROUP OF STATES AND SCATTERING CROSS-SECTION RESONANCE AND DECAY INTERACTION OF CHARGED PARTICLES AND RADIATION IN PERTURBATION THEORY ELECTRON IN AN ELECTROMAGNETIC FIELD: THE ABSORPTION CROSS-SECTION PHOTOELECTRIC EFFECT COULOMB EXCITATIONS OF AN ATOM IONIZATION THOMSON, RAYLEIGH, AND RAMAN SCATTERING IN SECOND-ORDER PERTURBATION SCATTERING IN ONE DIMENSION REFLECTION AND TRANSMISSION COEFFICIENTS INFINITE BARRIER FINITE BARRIER WITH INFINITE RANGE RIGID WALL PRECEDED BY A POTENTIAL WELL SQUARE-WELL POTENTIAL AND RESONANCES TUNNELING SCATTERING IN THREE DIMENSIONS – A FORMAL THEORY FORMAL SOLUTIONS IN TERMS OF GREEN’S FUNCTION LIPPMANN–SCHWINGER EQUATION BORN APPROXIMATION SCATTERING FROM A YUKAWA POTENTIAL RUTHERFORD SCATTERING CHARGE DISTRIBUTION PROBABILITY CONSERVATION AND THE OPTICAL THEOREM ABSORPTION RELATION BETWEEN THE T-MATRIX AND THE SCATTERING AMPLITUDE 20 10 THE S-MATRIX 20 11 UNITARITY OF THE S-MATRIX AND THE RELATION BETWEEN S AND T 20 12 PROPERTIES OF THE T-MATRIX AND THE OPTICAL THEOREM (AGAIN) 20 13 APPENDIX TO CHAPTER 20 14 PROBLEMS PARTIAL WAVE AMPLITUDES AND PHASE SHIFTS SCATTERING AMPLITUDE IN TERMS OF PHASE SHIFTS ?L, KL, AND TL INTEGRAL RELATIONS FOR ?L, KL, AND TL WRONSKIAN CALCULATION OF PHASE SHIFTS: SOME EXAMPLES ANALYTIC STRUCTURE OF THE S-MATRIX S-MATRIX POLES JOST FUNCTION FORMALISM LEVINSON’S THEOREM EXPLICIT CALCULATION OF THE JOST FUNCTION FOR L = INTEGRAL REPRESENTATION OF F0(K) POLES OF THE GREEN’S FUNCTION AND COMPOSITE SYSTEMS RELATION BETWEEN THE TIME-EVOLUTION OPERATOR AND THE GREEN’S FUNCTION STABLE AND UNSTABLE STATES SCATTERING AMPLITUDE AND RESONANCE COMPLEX POLES TWO TYPES OF RESONANCES THE REACTION MATRIX COMPOSITE SYSTEMS APPROXIMATION METHODS FOR BOUND STATES AND SCATTERING WKB APPROXIMATION VARIATIONAL METHOD EIKONAL APPROXIMATION LAGRANGIAN METHOD AND FEYNMAN PATH INTEGRALS EULER–LAGRANGE EQUATIONS N OSCILLATORS AND THE CONTINUUM LIMIT FEYNMAN PATH INTEGRALS ROTATIONS AND ANGULAR MOMENTUM ROTATION OF COORDINATE AXES SCALAR FUNCTIONS AND ORBITAL ANGULAR MOMENTUM TRANSFORMATION OF MATRIX ELEMENTS AND REPRESENTATIONS OF THE ROTATION OPERATOR GENERATORS OF INFINITESIMAL ROTATIONS: THEIR EIGENSTATES AND EIGENVALUES REPRESENTATIONS OF J 2 AND JI FOR J = 2 AND J = SPHERICAL HARMONICS SYMMETRY IN QUANTUM MECHANICS AND SYMMETRY GROUPS ROTATIONAL SYMMETRY PARITY TRANSFORMATION TIME REVERSAL SYMMETRY GROUPS DJ(R) FOR J = 2 AND J = 1: EXAMPLES OF SO(3) AND SU(2) GROUPS ADDITION OF ANGULAR MOMENTA COMBINING EIGENSTATES: SIMPLE EXAMPLES CLEBSCH–GORDAN COEFFICIENTS AND THEIR RECURSION RELATIONS COMBINING SPIN ½ AND ORBITAL ANGULAR MOMENTUM L IRREDUCIBLE TENSORS AND WIGNER–ECKART THEOREM IRREDUCIBLE SPHERICAL TENSORS AND THEIR PROPERTIES THE IRREDUCIBLE TENSORS: YLM(?, ?) AND DJ(?) WIGNER–ECKART THEOREM APPLICATIONS OF THE WIGNER–ECKART THEOREM APPENDIX TO CHAPTER 29: SO(3), SU(2) GROUPS AND YOUNG’S TABLEAU ENTANGLED STATES DEFINITION OF AN ENTANGLED STATE THE SINGLET STATE DIFFERENTIATING THE TWO APPROACHES BELL’S INEQUALITY SPECIAL THEORY OF RELATIVITY: KLEIN–GORDON AND MAXWELL’S EQUATIONS LORENTZ TRANSFORMATION CONTRAVARIANT AND COVARIANT VECTORS AN EXAMPLE OF A COVARIANT VECTOR GENERALIZATION TO ARBITRARY TENSORS RELATIVISTICALLY INVARIANT EQUATIONS KLEIN–GORDON AND MAXWELL’S EQUATIONS COVARIANT EQUATIONS IN QUANTUM MECHANICS KLEIN–GORDON EQUATIONS: FREE PARTICLES NORMALIZATION OF MATRIX ELEMENTS MAXWELL’S EQUATIONS PROPAGATORS VIRTUAL PARTICLES STATIC APPROXIMATION INTERACTION POTENTIAL IN NONRELATIVISTIC PROCESSES SCATTERING INTERPRETED AS AN EXCHANGE OF VIRTUAL PARTICLES 32 10 APPENDIX TO CHAPTER THE DIRAC EQUATION STANDARD REPRESENTATION AND SPINOR SOLUTIONS LARGE AND SMALL COMPONENTS OF U(P) SPIN ½ FOR THE DIRAC PARTICLE DIRAC EQUATION IN THE PRESENCE OF SPHERICALLY SYMMETRIC POTENTIALS SPIN–ORBIT COUPLING K-OPERATOR FOR THE SPHERICALLY SYMMETRIC POTENTIALS HYDROGEN ATOM RADIAL DIRAC EQUATION HYDROGEN ATOM STATES HYDROGEN ATOM WAVEFUNCTION DIRAC EQUATION IN A RELATIVISTICALLY INVARIANT FORM COVARIANT DIRAC EQUATION PROPERTIES OF THE ? -MATRICES CHARGE–CURRENT CONSERVATION IN A COVARIANT FORM SPINOR SOLUTIONS: UR(P) AND VR(P) NORMALIZATION AND COMPLETENESS CONDITION FOR UR(P) AND VR(P) GORDON DECOMPOSITION LORENTZ TRANSFORMATION OF THE DIRAC EQUATION INTERACTION OF A DIRAC PARTICLE WITH AN ELECTROMAGNETIC ?ELD CHARGED PARTICLE HAMILTONIAN DERIVING THE EQUATION ANOTHER WAY GORDON DECOMPOSITION AND ELECTROMAGNETIC CURRENT DIRAC EQUATION WITH EM FIELD AND COMPARISON WITH THE KLEIN–GORDON EQUATION PROPAGATORS: THE DIRAC PROPAGATOR SCATTERING MULTIPARTICLE SYSTEMS AND SECOND QUANTIZATION WAVEFUNCTIONS FOR IDENTICAL PARTICLES OCCUPATION NUMBER SPACE AND LADDER OPERATORS CREATION AND DESTRUCTION OPERATORS WRITING SINGLE-PARTICLE RELATIONS IN MULTIPARTICLE LANGUAGE: THE OPERATORS, N, H, AND P MATRIX ELEMENTS OF A POTENTIAL FREE FIELDS AND CONTINUOUS VARIABLES KLEIN–GORDON/SCALAR FIELD COMPLEX SCALAR FIELD DIRAC FIELD 37 10 MAXWELL FIELD 37 11 LORENTZ COVARIANCE FOR MAXWELL FIELD 37 12 PROPAGATORS AND TIME-ORDERED PRODUCTS 37 13 CANONICAL QUANTIZATION 37 14 CASIMIR EFFECT 37 15 PROBLEMS INTERACTIONS OF ELECTRONS AND PHONONS IN CONDENSED MATTER FERMI ENERGY INTERACTING ELECTRON GAS PHONONS ELECTRON–PHONON INTERACTION SUPERCONDUCTIVITY MANY-BODY SYSTEM OF HALF-INTEGER SPINS NORMAL STATES ( = 0, G ? =0) BCS STATES ( ? =0) BCS CONDENSATE IN GREEN’S FUNCTION FORMALISM MEISSNER EFFECT BOSE–EINSTEIN CONDENSATION AND SUPER?UIDITY MANY-BODY SYSTEM OF INTEGER SPINS SUPERFLUIDITY LAGRANGIAN FORMULATION OF CLASSICAL ?ELDS BASIC STRUCTURE NOETHER’S THEOREM EXAMPLES MAXWELL’S EQUATIONS AND CONSEQUENCES OF GAUGE INVARIANCE SPONTANEOUS SYMMETRY BREAKING BCS MECHANISM FERROMAGNETISM SSB FOR DISCRETE SYMMETRY IN CLASSICAL FIELD THEORY SSB FOR CONTINUOUS SYMMETRY NAMBU–GOLDSTONE BOSONS HIGGS MECHANISM BASIC QUANTUM ELECTRODYNAMICS AND FEYNMAN DIAGRAMS PERTURBATION THEORY FEYNMAN DIAGRAMS T(HI (X1) HI (X2)) AND WICK’S THEOREM FEYNMAN RULES CROSS-SECTION FOR 1 + 2 ?3 + BASIC TWO-BODY SCATTERING IN QED QED VS NONRELATIVISTIC LIMIT: ELECTRON–ELECTRON SYSTEM QED VS NONRELATIVISTIC LIMIT: ELECTRON–PHOTON SYSTEM RADIATIVE CORRECTIONS RADIATIVE CORRECTIONS AND RENORMALIZATION ELECTRON SELF-ENERGY ANOMALOUS MAGNETIC MOMENT AND LAMB SHIFT CALCULATING THE DIVERGENT INTEGRALS VERTEX FUNCTION AND THE MAGNETIC MOMENT CALCULATION OF THE VERTEX FUNCTION DIAGRAM DIVERGENT PART OF THE VERTEX FUNCTION RADIATIVE CORRECTIONS TO THE PHOTON PROPAGATOR DIVERGENT PART OF THE PHOTON PROPAGATOR MODIFICATION OF THE PHOTON PROPAGATOR AND PHOTON WAVEFUNCTION COMBINATION OF ALL THE DIVERGENT TERMS: BASIC RENORMALIZATION CONVERGENT PARTS OF THE RADIATIVE CORRECTIONS 45 10 APPENDIX TO CHAPTER BIBLIOGRAPHY