Physics 461 / Quantum Mechanics I P.E. Parris Department of Physics University of Missouri-Rolla Rolla, Missouri 65409 January 10, 2005 CONTENTS 1 Introduction 7 1.1WhatisQuantumMechanics?......................... 7 1.1.1 WhatisMechanics?.......................... 7 1.1.2 PostulatesofClassicalMechanics:.................. 8 1.2TheDevelopmentofWaveMechanics..................... 10 1.3TheWaveMechanicsofSchrödinger..................... 13 1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . 13 1.3.2 Schrödinger’sMechanicsforConservativeSystems......... 16 1.3.3 The Principle of Superposition and Spectral Decomposition . 17 1.3.4 TheFreeParticle............................ 19 1.3.5 Superpositions of Plane Waves and the Fourier Transform . 23 1.4Appendix:TheDeltaFunction........................ 26 2 The Formalism of Quantum Mechanics 31 2.1 Postulate I: SpecificationoftheDynamicalState.............. 31 2.1.1 PropertiesofLinearVectorSpaces.................. 31 2.1.2 Additional Definitions......................... 33 2.1.3 ContinuousBasesandContinuousSets................ 34 2.1.4 InnerProducts............................. 35 2.1.5 ExpansionofaVectoronanOrthonormalBasis.......... 38 2.1.6 Calculation of Inner Products Using an Orthonormal Basis . 39 2.1.7 ThePositionRepresentation..................... 40 2.1.8 TheWavevectorRepresentation.................... 41 2.2PostulateII:ObservablesofQuantumMechanicalSystems......... 43 2.2.1 OperatorsandTheirProperties.................... 43 2.2.2 MultiplicativeOperators........................ 45 2.2.3 DifferentialOperators......................... 47 2.2.4 Ket-BraOperators........................... 48 2.2.5 ProjectionOperators:Thecompletenessrelation.......... 49 2.2.6 MatrixElements............................ 51 2.2.7 Action of Operators on Bras of S∗ .................. 52 2.2.8 HermitianConjugation........................ 52 2.2.9 Hermitian,Anti-Hermitian,andUnitaryOperators......... 54 2.2.10MatrixRepresentationofOperators................. 55 2.2.11CanonicalCommutationRelations.................. 62 2.2.12 Matrix Elements of Unitary Operators (Changing Representation) 63 2.2.13RepresentationIndependentPropertiesofOperators........ 67 2.2.14EigenvaluesandEigenvectors..................... 69 2.2.15EigenpropertiesofHermitianOperators............... 70 2.2.16ObtainingEigenvectorsandEigenvalues............... 71 2.2.17CommonEigenstatesofCommutingObservables.......... 77 4 CONTENTS 2.3 Postulate III: The Measurement of Quantum Mechanical Systems . 80 2.3.1 Sum of Probabilities .......................... 85 2.3.2 MeanValues.............................. 86 2.3.3 StatisticalUncertainty......................... 87 2.3.4 TheUncertaintyPrinciple....................... 88 2.3.5 PreparationofaStateUsingaCSCO................ 90 2.4PostulateIV:Evolution............................ 91 2.4.1 Construction of the Hamiltonian and Other Observables . 93 2.4.2 SomeFeaturesofQuantumMechanicalEvolution.......... 94 2.4.3 EvolutionofMeanValues....................... 96 2.4.4 Eherenfest’sTheorem......................... 97 2.4.5 Evolution of Systems with Time Independent Hamiltonians . 99 2.4.6 TheEvolutionOperator........................ 102 3 The Harmonic Oscillator 105 3.1StatementoftheProblem........................... 105 3.1.1 Algebraic Approach to the Quantum Harmonic Oscillator . 107 3.1.2 Spectrum and Eigenstates of the Number Operator N ....... 110 3.1.3 TheEnergyBasis............................ 113 3.1.4 Action of Various Operators in the Energy Representation . 117 3.1.5 Time Evolution of the Harmonic Oscillator . ........... 121 4 Bound States of a Central Potential 123 4.1GeneralConsiderations............................. 123 4.2HydrogenicAtoms:TheCoulombProblem................. 127 4.3 The 3-D Isotropic Oscillator .......................... 131 5 Approximation Methods for Stationary States 133 5.1TheVariationalMethod............................ 133 5.2PerturbationTheoryforNondegenerateLevels............... 137 5.3PerturbationTheoryforDegenerateStates................. 146 5.3.1 Application: Stark Effect of the n =2Level of Hydrogen . 148 6 Many Particle Systems 151 6.1TheDirectProductofLinearVectorSpaces................. 151 6.1.1 Motion in 3 Dimensions Treated as a Direct Product of Vector Spaces155 6.1.2 TheStateSpaceofSpin-1/2Particles................ 156 6.2TheStateSpaceofManyParticleSystems.................. 156 6.3EvolutionofManyParticleSystems..................... 158 6.4SystemsofIdenticalParticles......................... 160 6.4.1 Construction of the Symmetric and Antisymmetric Subspaces . 162 6.4.2 NumberOperatorsandOccupationNumberStates......... 167 6.4.3 Evolution and Observables of a System of Identical Particles . 170 6.4.4 FockSpaceasaDirectSumofVectorSpaces............ 175 6.4.5 TheFockSpaceofIdenticalBosons................. 177 6.4.6 TheFockSpaceofIdenticalFermions................ 178 6.4.7 Observables of a System of Identical Particles Revisited . 182 6.4.8 FieldOperatorsandSecondQuantization.............. 186 CONTENTS 5 7 Angular Momentum and Rotations 189 7.1OrbitalAngularMomentumofOneorMoreParticles........... 189 7.2RotationofPhysicalSystems......................... 192 7.3RotationsinQuantumMechanics....................... 196 7.4CommutationRelationsforScalarandVectorOperators.......... 198 7.5RelationtoOrbitalAngularMomentum................... 200 7.6 Eigenstates and Eigenvalues of Angular Momentum Operators . 202 7.7OrthonormalizationofAngularMomentumEigenstates.......... 207 7.8OrbitalAngularMomentumRevisited.................... 211 7.9RotationalInvariance............................. 216 7.9.1 IrreducibleInvariantSubspaces.................... 216 7.9.2 RotationalInvarianceofStates.................... 220 7.9.3 RotationalInvarianceofOperators.................. 220 7.10AdditionofAngularMomenta......................... 221 7.11ReducibleandIrreducibleTensorOperators................. 230 7.12TensorCommutationRelations........................ 234 7.13TheWignerEckartTheorem......................... 235 8 Time Dependent Perturbations: Transition Theory 239 8.1GeneralConsiderations............................. 239 8.2PeriodicPerturbations:Fermi’sGoldenRule................ 245 8.3PerturbationsthatTurnOn.......................... 250 8.3.1 Sudden Perturbations ......................... 251 8.3.2 TheAdiabaticTheorem........................ 252 8.4Appendix:Landau-ZenerTransitions..................... 254 8.5 Free Particle Propagator . .......................... 259 8.6 Particle in a time dependent field....................... 260 9 Scattering Theory 263 9.1GeneralConsiderations............................. 263 9.2AnIntegralEquationfortheScatteringEigenfunctions........... 268 9.2.1 EvaluationOfTheGreen’sFunction................. 269 9.3TheBornExpansion.............................. 272 9.4ScatteringAmplitudesandT-Matrices.................... 272 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