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Quantum Physics (UCSD Physics 130)

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Quantum Physics (UCSD Physics 130) Quantum Physics (UCSD Physics 130) April 2, 2003 2 Contents 1 Course Summary 17 1.1 Problems with Classical Physics . .... 17 1.2 ThoughtExperimentsonDiffraction . ..... 17 1.3 Probability Amplitudes . 17 1.4 WavePacketsandUncertainty . ..... 18 1.5 Operators........................................ .. 19 1.6 ExpectationValues .................................. .. 19 1.7 Commutators ...................................... 20 1.8 TheSchr¨odingerEquation .. .. .. .. .. .. .. .. .. .. .. .. .... 20 1.9 Eigenfunctions, Eigenvalues and Vector Spaces . ......... 20 1.10 AParticleinaBox .................................... 22 1.11 Piecewise Constant Potentials in One Dimension . ...... 22 1.12 The Harmonic Oscillator in One Dimension . ... 24 1.13 Delta Function Potentials in One Dimension . .... 24 1.14 Harmonic Oscillator Solution with Operators . ...... 25 1.15 MoreFunwithOperators. .. .. .. .. .. .. .. .. .. .. .. .. .... 26 1.16 Two Particles in 3 Dimensions . .. 27 1.17 IdenticalParticles ................................. .... 28 1.18 Some 3D Problems Separable in Cartesian Coordinates . ........ 28 1.19 AngularMomentum.................................. .. 29 1.20 Solutions to the Radial Equation for Constant Potentials . .......... 30 1.21 Hydrogen........................................ .. 30 1.22 Solution of the 3D HO Problem in Spherical Coordinates . ....... 31 1.23 Matrix Representation of Operators and States . ........... 31 1.24 A Study of ℓ =1OperatorsandEigenfunctions . 32 1.25 Spin1/2andother2StateSystems . ...... 33 1.26 Quantum Mechanics in an Electromagnetic Field . ...... 33 1.27 Local Phase Symmetry in Quantum Mechanics and the Gauge Symmetry ...... 34 1.28 AdditionofAngularMomentum . ... 36 1.29 TimeIndependentPerturbationTheory . ........ 37 1.30 TheFineStructureofHydrogen . ...... 38 1.31HyperfineStructure ............................... ..... 39 1.32TheHeliumAtom ..................................... 40 1.33AtomicPhysics ..................................... .. 41 1.34Molecules ......................................... 42 1.35 TimeDependentPerturbationTheory . ....... 43 1.36RadiationinAtoms................................... .. 43 1.37 Classical Field Theory . ... 46 1.38 The Classical Electromagnetic Field . ..... 47 1.39 Quantization of the EM Field . ... 48 3 1.40ScatteringofPhotons .............................. ..... 50 1.41ElectronSelfEnergy ................................ .... 51 1.42TheDiracEquation .................................. .. 53 1.43TheDiracEquation .................................. .. 60 2 The Problems with Classical Physics 63 2.1 Black Body Radiation * .................................. 64 2.2 ThePhotoelectricEffect .............................. .... 69 2.3 The Rutherford Atom * .................................. 71 2.4 Atomic Spectra * ...................................... 73 2.4.1 The Bohr Atom * ................................. 75 2.5 DerivationsandComputations . .... 77 2.5.1 Black Body Radiation Formulas * ........................ 77 2.5.2 The Fine Structure Constant and the Coulomb Potential . ...... 77 2.6 Examples .......................................... 78 2.6.1 The Solar Temperature * ............................. 78 2.6.2 Black Body Radiation from the Early Universe * ................ 79 2.6.3 Compton Scattering * ............................... 79 2.6.4 Rutherford’s Nuclear Size * ............................ 81 2.7 SampleTestProblems ................................. .. 82 3 Diffraction 83 3.1 Diffraction from Two Slits . .. 83 3.2 Single Slit Diffraction . 86 3.3 DiffractionfromCrystals .............................. ... 86 3.4 TheDeBroglieWavelength .............................. .. 89 3.4.1 Computing DeBroglie Wavelengths . 90 3.5 Wave Particle Duality (Thought Experiments) . ...... 91 3.6 Examples .......................................... 95 3.6.1 Intensity Distribution for Two Slit Diffraction * ................ 95 3.6.2 Intensity Distribution for Single Slit Diffraction * ................ 95 3.7 SampleTestProblems ................................. .. 96 4 The Solution: Probability Amplitudes 97 4.1 DerivationsandComputations . .... 98 4.1.1 ReviewofComplexNumbers ........................... 98 4.1.2 Review of Traveling Waves . 98 4.2 SampleTestProblems ................................. .. 99 5 Wave Packets 100 5.1 Building a Localized Single-Particle Wave Packet . ..... 100 5.2 TwoExamplesofLocalizedWavePackets . ..... 101 5.3 The Heisenberg Uncertainty Principle . ..... 102 4 5.4 PositionSpaceandMomentumSpace . .... 103 5.5 Time Development of a Gaussian Wave Packet * .................... 104 5.6 DerivationsandComputations . .... 105 5.6.1 Fourier Series * ................................... 105 5.6.2 Fourier Transform * ................................ 106 5.6.3 IntegralofGaussian ............................... 107 5.6.4 Fourier Transform of Gaussian * ......................... 108 5.6.5 Time Dependence of a Gaussian Wave Packet * ................. 109 5.6.6 Numbers ...................................... 110 5.6.7 The Dirac Delta Function . 111 5.7 Examples .......................................... 112 5.7.1 TheSquareWavePacket ............................. 112 5.7.2 The Gaussian Wave Packet * ........................... 112 5.7.3 The Dirac Delta Function Wave Packet * .................... 113 5.7.4 CanI“See”insideanAtom............................ 113 5.7.5 Can I “See” inside a Nucleus . 113 5.7.6 EstimatetheHydrogenGroundStateEnergy . ..... 114 5.8 SampleTestProblems ................................. 115 6 Operators 117 6.1 OperatorsinPositionSpace .. .. .. .. .. .. .. .. .. .. .. .. .... 117 6.1.1 TheMomentumOperator ............................. 117 6.1.2 TheEnergyOperator ............................... 118 6.1.3 ThePositionOperator............................... 118 6.1.4 The Hamiltonian Operator . 118 6.2 OperatorsinMomentumSpace . .... 118 6.3 ExpectationValues.................................. .. 119 6.4 DiracBra-ketNotation............................... .... 120 6.5 Commutators...................................... 120 6.6 DerivationsandComputations . .... 121 6.6.1 VerifyMomentumOperator. 121 6.6.2 VerifyEnergyOperator ............................. 122 6.7 Examples .......................................... 122 6.7.1 Expectation Value of Momentum in a Given State . 122 6.7.2 Commutator of E and t .............................. 122 6.7.3 Commutator of E and x .............................. 123 6.7.4 Commutator of p and xn ............................. 123 6.7.5 Commutator of Lx and Ly ............................ 124 6.8 SampleTestProblems ................................. 124 5 7 The Schr¨odinger Equation 126 7.1 DerivingtheEquationfromOperators . ...... 126 7.2 The Flux of Probability * ................................. 127 7.3 TheSchr¨odingerWaveEquation . ..... 128 7.4 The Time IndependentSchr¨odingerEquation . ........ 128 7.5 DerivationsandComputations . .... 129 7.5.1 LinearOperators.................................. 129 7.5.2 Probability Conservation Equation * ....................... 130 7.6 Examples .......................................... 130 7.6.1 Solution to the Schr¨odinger Equation in a Constant Potential . ........ 130 7.7 SampleTestProblems ................................. 131 8 Eigenfunctions, Eigenvalues and Vector Spaces 132 8.1 EigenvalueEquations................................. .. 132 8.2 HermitianConjugateofanOperator . ...... 133 8.3 HermitianOperators................................. .. 134 8.4 EigenfunctionsandVectorSpace . ...... 135 8.5 TheParticleina1DBox ................................. 136 8.5.1 The Same Problem with Parity Symmetry . 138 8.6 MomentumEigenfunctions ............................. .. 139 8.7 DerivationsandComputations . .... 140 8.7.1 Eigenfunctions of Hermitian Operators are Orthogonal . ........ 140 8.7.2 Continuity of Wavefunctions and Derivatives . .... 141 8.8 Examples .......................................... 142 8.8.1 HermitianConjugateofaConstantOperator . ..... 142 ∂ 8.8.2 Hermitian Conjugate of ∂x ............................ 142 8.9 SampleTestProblems ................................. 142 9 One Dimensional Potentials 145 9.1 Piecewise Constant Potentials in 1D . .... 145 9.1.1 The General Solution for a Constant Potential . ..... 145 9.1.2 ThePotentialStep................................. 145 9.1.3 The Potential Well with E > 0 * ......................... 147 9.1.4 Bound States in a Potential Well * ........................ 149 9.1.5 ThePotentialBarrier ............................... 151 9.2 The 1D Harmonic Oscillator . 153 9.3 The Delta Function Potential * .............................. 155 9.4 The Delta Function Model of a Molecule * ........................ 156 9.5 The Delta Function Model of a Crystal * ........................ 157 9.6 TheQuantumRotor .................................. 159 9.7 DerivationsandComputations . .... 160 9.7.1 Probability Flux for the Potential Step * .................... 160 6 9.7.2 Scattering from a 1D Potential Well * ...................... 160 9.7.3 Bound States of a 1D Potential Well * ...................... 162 9.7.4 Solving the HO Differential Equation * ..................... 164 9.7.5 1D Model of a Molecule Derivation * ....................... 167 9.7.6 1D Model of a Crystal Derivation * ....................... 167 9.8 Examples .......................................... 169 9.9 SampleTestProblems ................................. 170 10 Harmonic Oscillator Solution using Operators 172 10.1 Introducing A and A† ..................................
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