LECTURE NOTES ON QUANTUM MECHANICS Dr. Shun-Qing Shen Department of Physics The University of Hong Kong September 2004 Con ten ts 0.1 GeneralInformation ix 1 Fundamental Concepts 1 1.1 Relation between experimental interpretations and theoretical inferences . . . 2 1.2 MaterialsfromBritannicaOnline 3 1.2.1 Photoelectric eect 3 1.2.2 Frank-Hertzexperiment 5 1.2.3 Compton eect 6 1.3 TheStern-GerlachExperiment 7 1.3.1 TheStern-Gerlachexperiment 7 1.3.2 SequentialStern-GerlachExperiment 9 1.3.3 AnalogywithPolarizationofLight 11 1.4 DiracNotationandOperators 11 1.5 BaseketsandMatrixRepresentation 14 1.5.1 EigenketsofanObservable 14 1.5.2 EigenketsasBasekets: 15 1.5.3 MatrixRepresentation: 16 1.6 Measurements,Observables&TheUncertaintyRelation 18 ii CONTENTS – MANUSCRIPT 1.6.1 Measurements 18 1.6.2 Spin1/2system 19 1.6.3 ProbabilityPostulate 20 1.6.4 S { and S | 21 1.6.5 TheAlgebraofSpinOperators 23 1.6.6 Observable 24 1.7 ChangeofBasis 26 1.7.1 Transformation Operator 27 1.7.2 TransformationMatrix 28 1.7.3 Diagonalization 30 1.8 Position,Momentum,andTranslation 31 1.8.1 ContinuousSpectra 31 1.8.2 Some properties of the function 32 1.8.3 PositionEigenketsandPositionMeasurements 33 1.8.4 Translation 34 1.9 TheUncertaintyRelation 38 2 Quantum Dynamics 46 2.1 TimeEvolutionandtheSchrödingerEquation 46 2.1.1 TimeEvolutionOperator 46 2.1.2 TheSchrodingerEquation. 48 2.1.3 TimeDependenceofExpectationValue:SpinPrecession. 52 2.2 TheSchrodingerversustheHeisenbergPicture 54 2.2.1 Unitaryoperators 54 2.2.2 TwoApproaches 55 iii CONTENTS – MANUSCRIPT 2.2.3 TheHeisenbergEquationofMotion 56 2.2.4 HowtoconstructaHamiltonian 57 2.3 SimpleHarmonicOscillator 59 2.3.1 Eigenvalueandeigenstates 59 2.3.2 Time Development of the Oscillator . . 66 2.3.3 TheCoherentState 67 2.4 SchrodingerWaveEquation:SimpleHarmonicOscillator 68 2.5 Propagators and Feynman Path Integrals . . . 70 2.5.1 Propagators in Wave Mechanics. 70 2.5.2 Propagator as a Transition Amplitude. . 75 2.6 TheGaugeTransformationandPhaseofWaveFunction 79 2.6.1 ConstantPotential 79 2.6.2 GaugeTransformationinElectromagnetism 81 2.6.3 TheGaugeTransformation 84 2.6.4 The Aharonov-Bohm Eect 86 2.7 InterpretationofWaveFunction. 88 2.7.1 What’s   ({)? 88 2.7.2 TheClassicalLimit 90 2.8 Examples 91 2.8.1 Onedimensionalsquarewellpotential 91 2.8.2 A charged particle in a uniform magnetic field . . . 95 3 Theory of Angular Momentum 98 3.1 RotationandAngularMomentum 98 3.1.1 Finite versus infinitesimal rotation . . . 99 iv CONTENTS – MANUSCRIPT 3.1.2 Orbitalangularmomentum 104 3.1.3 Rotationoperatorforspin1/2 106 3.1.4 Spin precession revisited . 107 3.2 RotationGroupandtheEulerAngles 108 3.2.1 TheGroupConcept 108 3.2.2 Orthogonal Group 109 3.2.3 “Special”? 110 3.2.4 UnitaryUnimodularGroup 110 3.2.5 EulerRotations 112 3.3 EigenvaluesandEigenketsofAngularMomentum 114 3.3.1 RepresentationofRotationOperator 120 3.4 SchwingerOscillatorModel 121 3.4.1 Spin1/2system 125 3.4.2 Two-spin—1/2 system . . 126 3.4.3 ExplicitFormulaforRotationMatrices. 128 3.5 Combination of Angular Momentum and Clebsh-Gordan Coe!cients 130 3.5.1 Clebsch-Gordan coe!cients 133 3.6 Examples 137 3.6.1 Twospin-1/2systems 137 3.6.2 Spin-orbitcoupling 137 4 Symmetries in Physics 140 4.1 SymmetriesandConservationLaws 141 4.1.1 SymmetryinClassicalPhysics 141 4.1.2 SymmetryinQuantumMechanics 142 v CONTENTS – MANUSCRIPT 4.1.3 Degeneracy 143 4.1.4 Symmetryandsymmetrybreaking 144 4.1.5 Summary:symmetriesinphysics 147 4.2 DiscreteSymmetries 148 4.2.1 Parity 148 4.2.2 TheMomentumOperator 150 4.2.3 TheAngularMomentum 151 4.2.4 LatticeTranslation 154 4.3 PermutationSymmetryandIdenticalParticles 157 4.3.1 Identicalparticles 157 4.4 TimeReversal 162 4.4.1 Classicalcases 162 4.4.2 Antilinear Operators . . . 162 4.4.3 Antiunitaryoperators 162 4.4.4 Tforazerospinparticle 162 4.4.5 Tforanonzerospinparticle 162 5 Approximation Methods for Bound States 163 5.1 TheVariationMethod 163 5.1.1 Expectationvalueoftheenergy 164 5.1.2 Particle in a one-dimensional infinite square well . . 165 5.1.3 GroundStateofHeliumAtom 166 5.2 StationaryPerturbationTheory:NondegenerateCase 168 5.2.1 StatementoftheProblem 168 5.2.2 TheTwo-StateProblem 169 vi CONTENTS – MANUSCRIPT 5.2.3 Formal Development of Perturbation··· 170 5.3 ApplicationofthePerturbationExpansion 174 5.3.1 Simpleharmonicoscillator 174 5.3.2 Atomichydrogen 177 5.4 StationaryPerturbationTheory: DegenerateCase 179 5.4.1 Revisitedtwo-stateproblem 180 5.4.2 Thebasicprocedureofdegenerateperturbationtheory 181 5.4.3 Example: Zeeman Eect 183 5.4.4 Example: First Order Stark EectinHydrogen 184 5.5 TheWentzel-Kramers-Brillouin(WKB)approximation 186 5.6 Time-dependent Problem: Interacting Picture and Two-State Problem . . . . 186 5.6.1 Time-dependentPotentialandInteractingPicture 187 5.6.2 Time-dependentTwo-StateProblem 188 5.7 Time-dependentPerturbationProblem 191 5.7.1 PerturbationTheory 191 5.7.2 Time-independentperturbation 193 5.7.3 Harmonicperturbation 193 5.7.4 TheGoldenRule 194 6 Collision Theory 196 6.1 Collisions in one- and three-dimensions 197 6.1.1 One-dimensionalsquarepotentialbarriers 197 6.2 Collision in three dimensions . . 200 6.3 ScatteringbySphericallySymmetricPotentials 204 6.4 Applications 209 vii CONTENTS – MANUSCRIPT 6.4.1 Scatteringbyasquarewell 209 6.4.2 Scatteringbyahard-spherepotential 211 6.5 Approximate Collision Theory . . 212 6.5.1 TheLippman-SchwingerEquation 212 6.5.2 TheBornApproximation 216 6.5.3 Application:fromYukawapotentialtoColoumbpotential 217 6.5.4 IdenticalParticlesandScattering 218 6.6 Landau-ZenerProblem 219 7SelectedTopics 220 7.1 QuantumStatistics 220 7.1.1 DensityOperatorandEnsembles 220 7.1.2 QuantumStatisticalMechanism 223 7.1.3 QuantumStatistics 226 7.1.4 Systemsofnon-interactionparticles 227 7.1.5 Bose-EinsteinCondensation 230 7.1.6 Freefermiongas 232 7.2 Quantum Hall Eect 233 7.2.1 Hall Eect 234 7.2.2 Quantum Hall Eect 236 7.2.3 Laughlin’sTheory 238 7.2.4 Charged particle in the presence of a magnetic field 240 7.2.5 Landau Level and Quantum Hall Eect 243 7.3 QuantumMagnetism 244 7.3.1 SpinExchange 245 viii CONTENTS – MANUSCRIPT 7.3.2 Two-SiteProblem 247 7.3.3 Ferromagnetic Exchange (M?0) 249 7.3.4 AntiferromagneticExchange 251 0.1 General Information • Aim/Following-up: The course provides an introduction to advanced techniques in quantum mechanics and their application to several selected topics in condensed matter physics. • Contents: Dirac notation and formalism, time evolution of quantum systems, angu- lar momentum theory, creation and annihilation operators (the second quantization representation), symmetries and conservation laws, permutation symmetry and identi- cal particles, quantum statistics, non-degenerate and degenerate perturbation theory, time-dependent perturbation theory, the variational method • Prerequisites: PHYS2321, PHYS2322, PHYS2323, and PHYS2325 • Co-requisite:Nil • Teaching: 36 hours of lectures and tutorial classes • Duration: One semester (1st semester) • As sessment: One-three hour examination (70%) and course assessment (30%) • Textb ook: J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994) • Web page: http://bohr.physics.hku.hk/~phys3332/ All lectures notes in pdf files can be download from the site. ix CONTENTS – MANUSCRIPT • References:L.Schi, Quantum Mechanics (McGraw-Hill, 1968, 3nd ed.); Richard Feynman, Robert B. Leighton, and Matthew L. Sands, Feynman Lectures on Physics Vol. III, (Addison-Wesley Publishing Co., 1965); L. D. Landau and E. M. Lifshitz, Quantum Mechanics Time and Venue: Year 2004 (Weeks 2 — 14) 11:40 -12:30: Monday/S802 11:40 -12:30: Wednesday/S802 11:40 -12:30: Friday/S802 x [...]... book) In this section we formulate the basic mathematics of vector spaces as used in quantum mechanics The theory of linear algebra has been known to mathematician before the birth of quantum mechanics, but the Dirac notation has many advantages At the early time of quantum mechanics this notation was used to unify Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics The notation in this course... photons transfer some of their energy and momentum to the electrons, which in turn recoil In the instant of the collision, new photons of less energy and momentum are produced that scatter at angles the size of which depends on the amount of energy lost to the recoiling electrons Because of the relation between energy and wavelength, the scattered photons have a longer wavelength that also depends on. .. called photons Photons have energy and momentum just as material particles do; they also have wave characteristics, such as wavelength and frequency The energy of photons is directly proportional to their frequency and inversely proportional to their wavelength, so lower-energy photons have lower frequencies and longer wavelengths In the Compton e ect, individual photons collide with single electrons that... The angle is about = = 2 = 2 They will reach at di erent places on the screen Classically: all values of between | | and = cos (0 ) would be expected to realize | | It has a continuous distribution Experimentally: only two values of z component of are observed (electron spin ) Consequence: the spin of electron has two discrete values along the magnetic 8 CHAPTER 1 – MANUSCRIPT Figure 1.2: Beam from... elastically scattered by electrons; it is a principal way in which radiant energy is absorbed in matter The e ect has proved to be one of the cornerstones of quantum mechanics, which accounts for both wave and particle properties of radiation as well as of matter The American physicist Arthur Holly Compton explained (1922; published 1923) the wavelength increase by considering X rays as composed of... (reciprocal fine structure constant) = 0 927 × 10 2 2 20 erg/oersted (Bohr magneton) = 5 11 × 105 eV (electron rest energy) = 938MeV (proton rest energy) 1eV = 1 602 × 10 12 erg 1 eV/c = 12 400Å 1eV = 11 600K xi Chapter 1 Fundamental Concepts At the present stage of human knowledge, quantum mechanics can be regarded as the fundamental theory of atomic phenomena The experimental data on which it is based are... are derived from physical events that almost entirely beyond the range of direct human perception It is not surprising that the theory embodies physical concepts that are foreign to common daily experience The most traditional way to introduce the quantum mechanics is to follow the historical development of theory and experiment — Planck’s radiation law, the EinsteinDebye’s theory of specific heat, the... electrons ejected from a material is equal to the frequency of the incident light times Planck’s constant, less the work function The resulting photoelectric equation of Einstein can be expressed by energy of the ejected electron, Planck’s constant, = , in which is the maximum kinetic is a constant, later shown to be numerically the same as is the frequency of the incident light, and is the work function... of the temperature and a function of the incident photon’s energy The equation is = 2 ( ), in which I is the photoelectric current, a and A are constants, and ( ) is an exponential series, whose numerical values have been tabulated; the dimensionless value x equals the kinetic energy of the emitted electrons divided by the product of the temperature and the Boltzmann constant of the kinetic theory:... Ket space: In quantum mechanics, a physical state, for example, a silver atom with definite spin orientation, is represented by a state vector in a complex vector space, denoted by | i, a ket The state ket is postulated to contain complete information about physical state The dimensionality of a complex vector space is specified according to the nature of physical system under consideration An observable . QuantumStatistics 220 7.1.1 DensityOperatorandEnsembles 220 7.1.2 QuantumStatisticalMechanism 223 7.1.3 QuantumStatistics 226 7.1.4 Systemsofnon-interactionparticles 227 7.1.5 Bose-EinsteinCondensation. 137 4 Symmetries in Physics 140 4.1 SymmetriesandConservationLaws 141 4.1.1 SymmetryinClassicalPhysics 141 4.1.2 SymmetryinQuantumMechanics 142 v CONTENTS – MANUSCRIPT 4.1.3 Degeneracy 143 4.1.4 Symmetryandsymmetrybreaking. PHYS2321, PHYS2322, PHYS2323, and PHYS2325 • Co-requisite:Nil • Teaching: 36 hours of lectures and tutorial classes • Duration: One semester (1st semester) • As sessment: One-three hour examination (70%)