Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction ................................... 39 4.2 Schr¨odinger’sEquation . 39 4.3 NormalizationoftheWavefunction . 39 4.4 ExpectationValuesandVariances . 42 4.5 Ehrenfest’sTheorem.............................. 43 4.6 Operators .................................... 45 4.7 MomentumRepresentation. 47 4.8 Heisenberg’sUncertaintyPrinciple . ....... 50 4.9 EigenstatesandEigenvalues . 53 4.10 Measurement .................................. 56 4.11 ContinuousEigenvalues. 57 4.12 StationaryStates ............................... 60 5 One-Dimensional Potentials 63 5.1 Introduction ................................... 63 5.2 InfinitePotentialWell . .. .. .. .. .. .. .. .. .. .. .. .. .. 63 5.3 SquarePotentialBarrier. 64 5.4 WKBApproximation .............................. 70 5.5 ColdEmission .................................. 72 5.6 AlphaDecay ................................... 73 5.7 SquarePotentialWell ............................. 75 5.8 SimpleHarmonicOscillator . 80 6 Multi-Particle Systems 85 6.1 Introduction ................................... 85 6.2 FundamentalConcepts ............................. 85 6.3 Non-InteractingParticles . 86 6.4 Two-ParticleSystems ............................. 88 6.5 IdenticalParticles .............................. 89 7 Three-Dimensional Quantum Mechanics 93 7.1 Introduction ................................... 93 7.2 FundamentalConcepts ............................. 93 7.3 ParticleinaBox................................. 96 7.4 DegenerateElectronGases . 97 7.5 White-DwarfStars................................ 100 CONTENTS 3 8 Orbital Angular Momentum 103 8.1 Introduction ...................................103 8.2 AngularMomentumOperators. 103 8.3 RepresentationofAngularMomentum . 105 8.4 EigenstatesofAngularMomentum. 106 8.5 Eigenvalues of Lz ................................107 8.6 Eigenvalues of L2 ................................108 8.7 SphericalHarmonics .............................. 110 9 Central Potentials 115 9.1 Introduction ...................................115 9.2 DerivationofRadialEquation . 115 9.3 InfiniteSphericalPotentialWell . 118 9.4 HydrogenAtom .................................121 9.5 RydbergFormula ................................126 10 Spin Angular Momentum 129 10.1 Introduction ................................... 129 10.2 SpinOperators.................................. 129 10.3 SpinSpace....................................130 2 10.4 Eigenstates of Sz and S .............................131 10.5 PauliRepresentation . 133 10.6 SpinPrecession ................................. 136 11 Addition of Angular Momentum 141 11.1 Introduction ................................... 141 11.2 GeneralPrinciples. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 141 11.3 AngularMomentumintheHydrogenAtom . 144 11.4 TwoSpinOne-HalfParticles . 147 12 Time-Independent Perturbation Theory 151 12.1 Introduction ................................... 151 12.2 ImprovedNotation ............................... 151 12.3 Two-StateSystem ................................ 153 12.4 Non-DegeneratePerturbationTheory . 155 12.5 QuadraticStarkEffect. 156 12.6 DegeneratePerturbationTheory . 160 12.7 LinearStarkEffect. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 163 12.8 FineStructureofHydrogen. 164 12.9 ZeemanEffect ..................................169 12.10HyperfineStructure . 170 13 Time-Dependent Perturbation Theory 175 4 QUANTUM MECHANICS 13.1 Introduction ................................... 175 13.2 PreliminaryAnalysis. 175 13.3 Two-StateSystem ................................ 177 13.4 SpinMagneticResonance. 178 13.5 PerturbationExpansion . 180 13.6 HarmonicPerturbations. 181 13.7 ElectromagneticRadiation . 183 13.8 ElectricDipoleApproximation . 186 13.9 SpontaneousEmission . 188 13.10 RadiationfromaHarmonicOscillator . .......190 13.11SelectionRules ................................ 191 13.12 2P 1S TransitionsinHydrogen . 192 13.13IntensityRules................................ 194 13.14ForbiddenTransitions→ . 194 14 Variational Methods 197 14.1 Introduction ................................... 197 14.2 VariationalPrinciple. 197 14.3 HeliumAtom ..................................199 14.4 HydrogenMoleculeIon. 204 15 Scattering Theory 209 15.1 Introduction ................................... 209 15.2 Fundamentals ..................................209 15.3 BornApproximation.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 211 15.4 PartialWaves ..................................213 15.5 DeterminationofPhase-Shifts . 216 15.6 HardSphereScattering . 217 15.7 LowEnergyScattering . 219 15.8 Resonances....................................220 Introduction 5 1 Introduction 1.1 Intended audience These lecture notes outline a single semester course on non-relativistic quantum mechanics which is primarily intended for upper-division undergraduate physics majors. The course assumes some previous knowledge of physics and mathematics. In particular, prospective students should be reasonably familiar with Newtonian dynamics, elementary classical electromagnetism and special relativity, the physics and mathematics of waves (includ- ing the representation of waves via complex functions), basic probability theory, ordinary and partial differential equations, linear algebra, vector algebra, and Fourier series and transforms. 1.2 Major Sources The textbooks which I have consulted most frequently whilst developing course material are: The Principles of Quantum Mechanics, P.A.M. Dirac, 4th Edition (revised), (Oxford Univer- sity Press, Oxford UK, 1958). Quantum Mechanics, E. Merzbacher, 2nd Edition, (John Wiley & Sons, New York NY, 1970). Introduction to the Quantum Theory, D. Park, 2nd Edition, (McGraw-Hill, New York NY, 1974). Modern Quantum Mechanics, J.J. Sakurai, (Benjamin/Cummings, Menlo Park CA, 1985). Quantum Theory, D. Bohm, (Dover, New York NY, 1989). Problems in Quantum Mechanics, G.L. Squires, (Cambridge University Press, Cambridge UK, 1995). Quantum Physics, S. Gasiorowicz, 2nd Edition, (John Wiley & Sons, New York NY, 1996). Nonclassical Physics, R. Harris, (Addison-Wesley, Menlo Park CA, 1998). Introduction to Quantum Mechanics, D.J. Griffiths, 2nd Edition, (Pearson Prentice Hall, Upper Saddle River NJ, 2005). 6 QUANTUM MECHANICS 1.3 Aim of Course The aim of this course is to develop non-relativistic quantum mechanics as a complete theory of microscopic dynamics, capable of making detailed predictions, with a minimum of abstract mathematics. 1.4 Outline of Course The first part of the course is devoted to an in-depth exploration of the basic principles of quantum mechanics. After a brief review of probability theory, in Chapter 2, we shall start, in Chapter 3, by examining how many of the central ideas of quantum mechanics are a direct consequence of wave-particle duality—i.e., the concept that waves sometimes act as particles, and particles as waves. We shall then proceed to investigate the rules of quantum mechanics in a more systematic fashion in Chapter 4. Quantum mechanics is used to examine the motion of a single particle in one dimension, many particles in one dimension, and a single particle in three dimensions, in Chapters 5, 6, and 7, respectively. Chapter 8 is devoted to the investigation of orbital angular momentum, and Chapter 9 to the closely related subject of particle motion in a central potential. Finally, in Chapters 10 and 11, we shall examine spin angular momentum, and the addition of orbital and spin angular momentum, respectively. The second part of this course describes selected practical applications of quantum mechanics. In Chapter 12, time-independent perturbation theory is used to investigate the Stark effect, the Zeeman effect, fine structure, and hyperfine structure, in the hydrogen atom. Time-dependent perturbation theory is employed to study radiative transitions in the hydrogen atom in Chapter 13. Chapter 14 illustrates the use of variational methods in quantum mechanics. Finally, Chapter 15 contains an introduction to quantum scattering theory. Probability Theory 7 2 Probability Theory 2.1 Introduction This section is devoted