Building Blocks of Quantum Mechanics: Theory and Applications

265 Pages·2022·10.704 MB·English

by  Tao Xiang

Building Blocks of
Quantum Mechanics

Building Blocks of
Quantum Mechanics
Theory and Applications
Tao Xiang
First edition published 2022
by CRC Press
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© 2022 Tao Xiang
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ISBN: 978-1-032-00610-9 (hbk)
ISBN: 978-0-367-77150-8 (pbk)
ISBN: 978-1-003-17488-2 (ebk)
DOI: 10.1201/9781003174882
Typeset in Nimbus Roman
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Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Dedication
Tomymother.

Contents
Preface.....................................................................................................................xiii
Notations..................................................................................................................xv
FormulasinSIunitsandGaussianunits................................................................xvii
Tableoffundamentalconstants..............................................................................xix
Chapter1 Introduction.....................................................................................1
1.1 Briefhistoryofquantummechanics.......................................1
1.2 Schro¨dingerequation..............................................................4
1.3 Probabilityinterpretationofwavefunction............................6
1.4 StationarySchro¨dingerequation.............................................8
1.5 Conservationofprobability....................................................8
1.6 Quantumsuperposition.........................................................10
1.6.1 Nocloningtheorem..................................................11
1.6.2 Schro¨dingercat........................................................12
1.7 Operators...............................................................................12
1.8 Quantummeasurement.........................................................13
1.8.1 Stern-Gerlachexperiment........................................14
1.9 Expectationvalues................................................................16
1.10 Problems...............................................................................17
Chapter2 One-dimensionalEigen-problem...................................................19
2.1 Symmetricpotentialandparity.............................................19
2.2 Freeparticle..........................................................................20
2.3 Delta-functionnormalization................................................23
2.4 Infinitesquarewellpotential ...............................................24
2.5 Finitesquarewellpotential..................................................27
2.5.1 Boundstates−V <E≤0........................................27
2.5.2 ScatteringstatesE>0.............................................31
2.6 Quantumtunneling...............................................................33
2.7 Delta-functionpotential........................................................35
2.7.1 Boundstate(α<0andE<0)...............................36
2.7.2 Scatteringstate(E>0)............................................37
2.8 TheWKBapproximation.....................................................38
2.8.1 Solutionaroundaturningpoint...............................40
vii
viii Contents
2.8.2 Theconnectionformulae.........................................41
2.8.3 Quantizationofenergylevels...................................43
2.9 Problems...............................................................................45
Chapter3 Representationtheoryofquantumstates.......................................47
3.1 Representation......................................................................47
3.1.1 Diracbracketnotations............................................47
3.1.2 Representationofquantumstates............................48
3.1.3 Hermitianoperators.................................................49
3.1.4 EigenstatesofHermitianoperators..........................50
3.1.5 Representationofoperators.....................................53
3.1.6 RepresentationofSchro¨dingerequation..................53
3.1.7 Feynman-Hellmanntheorem...................................53
3.2 Basistransformation.............................................................54
3.2.1 Example:Fromrealtomomentumspacerepre-
sentation...................................................................55
3.3 Commutators.........................................................................56
3.3.1 Propertiesofcommutableoperators........................56
3.3.2 Propertiesofnoncommutableoperators..................57
3.4 Schro¨dingerpicture...............................................................59
3.4.1 Virialtheorem..........................................................60
3.4.2 Ehrenfesttheorem....................................................60
3.5 Heisenbergpicture................................................................61
3.6 Uncertaintyprinciple............................................................62
3.7 Thetime-energyuncertaintyprinciple..................................64
3.8 Problems...............................................................................66
Chapter4 HarmonicOscillators.....................................................................69
4.1 One-dimensionalharmonicoscillator...................................69
4.1.1 Ladderoperators......................................................69
4.1.2 Eigen-spectrum........................................................70
4.1.3 Eigenfunction...........................................................71
4.1.4 Occupationrepresentation.......................................72
4.2 Coherentstate.......................................................................73
4.2.1 Minimumuncertaintystate......................................73
4.2.2 Wavefunctionofthecoherentstate.........................74
4.3 Chargedparticlesinanelectromagneticfield.......................75
4.3.1 Minimalcoupling.....................................................75
4.3.2 Gaugeinvariance......................................................76
4.3.3 Probabilitycurrent...................................................76
4.3.4 Aharonov-Bohmeffect............................................77
4.4 Landaulevels........................................................................79
4.4.1 Landaugauge...........................................................79
4.4.2 DegeneracyofLandaulevels...................................80
Contents ix
4.4.3 Symmetricgauge.....................................................81
4.4.4 LowestLandaulevel................................................82
4.5 Problems...............................................................................85
Chapter5 AngularMomentum......................................................................87
5.1 Orbitalangularmomentum...................................................87
5.2 Generalangularmomentum.................................................88
5.2.1 Matrix representation of angular momentum
operators...................................................................90
5.3 Eigenfunctionsoforbitalangularmomentum......................91
5.4 Spinangularmomentum.......................................................95
5.4.1 Paulimatrices...........................................................95
5.4.2 EigenstatesofS=1/2.............................................96
5.4.3 QubitandBlochsphere............................................97
5.5 Additionoftwoangularmomenta........................................98
5.5.1 Clebsch-Gordancoefficients..................................100
5.5.2 AdditionoftwoS=1/2spins...............................101
∗
5.6 Wigner-Eckarttheorem .....................................................102
5.6.1 ProofoftheWigner-Eckarttheorem......................103
5.7 Problems.............................................................................105
Chapter6 Centralpotential..........................................................................107
6.1 Three-dimensionalpotentialwithsphericalsymmetry.......107
6.2 Hydrogenicatom................................................................109
6.2.1 Hamiltonianinthecenter-of-massframework......109
6.2.2 Boundstatesolutions.............................................110
6.2.3 Solutionsbyseriesexpansion................................111
6.2.4 Radialwavefunction.............................................113
6.2.5 Rydbergformula....................................................115
6.3 Partialwavemethod............................................................116
6.3.1 Partialwaveexpansion...........................................117
6.3.2 Scatteringamplitude..............................................119
6.3.3 Scatteringcrosssection..........................................120
6.3.4 Hard-spherescattering...........................................120
∗
6.4 Supersymmetricquantummechanicsapproach ................121
6.4.1 Supersymmetricsolutionofthehydrogenicatom.125
6.5 Problems.............................................................................128
Chapter7 IdenticalParticles........................................................................131
7.1 Permutationsymmetry........................................................131
7.2 Bose-EinsteinandFermi-Diracstatistics............................132
7.2.1 Exchangedegeneracy.............................................132
7.2.2 Anti-symmetrizedwavefunctions.........................133