Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics

228 Pages·2005·5.32 MB·english

by  C. L. Tang

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FundamentalsofQuantumMechanics
Quantummechanicshasevolvedfromasubjectofstudyinpurephysicstoonewitha
wide range of applications in many diverse fields. The basic concepts of quantum
mechanics are explained in this book in a concise and easy-to-read manner, leading
towardapplicationsinsolidstateelectronicsandmodernoptics.Followingalogical
sequence,thebookisfocusedonthekeyideasandisconceptuallyandmathematically
self-contained. The fundamental principles of quantum mechanics are illustrated by
showingtheirapplicationtosystemssuchasthehydrogenatom,multi-electronions
andatoms,theformationofsimpleorganicmoleculesandcrystallinesolidsofprac-
tical importance. It leads on from these basic concepts to discuss some of the most
importantapplicationsinmodernsemiconductorelectronicsandoptics.
Containingmanyhomeworkproblems,thebookissuitableforsenior-levelunder-
graduateandgraduatelevelstudentsinelectricalengineering,materialsscience,and
appliedphysicsandchemistry.
C. L. Tang is the Spencer T. Olin Professor of Engineering at Cornell University,
Ithaca,NY.Hisresearchinteresthasbeeninquantumelectronics,nonlinearoptics,
femtosecondopticsandultrafastprocessinmoleculesandsemiconductors,andhehas
publishedextensivelyinthesefields.HeisaFellowoftheIEEE,theOpticalSocietyof
America, and the Americal Physical Society, and is a member of the US National
AcademyofEngineering.HewasthewinneroftheCharlesH.TownesAwardofthe
OpticalSocietyofAmericain1996.

Fundamentals of Quantum
Mechanics
For Solid State Electronics and Optics
C. L. TANG
CornellUniversity,Ithaca,NY
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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The Edinburgh Building, Cambridge , UK
Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2005
This publication is in copyright. Subject to statutory exception and to the provision of
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without the written permission of Cambridge University Press.
First published in print format 2005
isbn-13 978-0-511-12595-9
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isbn-10 0-511-12595-x
eBook (NetLibrary)
isbn-13 978-0-521-82952-6
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To
Louise

Contents
Preface pagex
1 Classicalmechanicsvs.quantummechanics 1
1.1 Briefoverviewofclassicalmechanics 1
1.2 Overviewofquantummechanics 2
2 Basicpostulatesandmathematicaltools 8
2.1 Statefunctions(Postulate1) 8
2.2 Operators(Postulate2) 12
2.3 Equationsofmotion(Postulate3) 18
2.4 Eigenfunctions,basisstates,andrepresentations 21
2.5 Alternativenotationsandformulations 23
2.6 Problems 31
3 Wave/particledualityanddeBrogliewaves 33
3.1 FreeparticlesanddeBrogliewaves 33
3.2 Momentumrepresentationandwavepackets 37
3.3 Problems 39
4 Particlesatboundaries,potentialsteps,barriers,andinquantumwells 40
4.1 Boundaryconditionsandprobabilitycurrents 40
4.2 Particlesatapotentialstep,upordown 43
4.3 Particlesatabarrierandthequantummechanicaltunnelingeffect 47
4.4 Quantumwellsandboundstates 50
4.5 Three-dimensionalpotentialboxorquantumwell 59
4.6 Problems 60
5 Theharmonicoscillatorandphotons 63
5.1 TheharmonicoscillatorbasedonHeisenberg’sformulationofquantum
mechanics 63
5.2 TheharmonicoscillatorbasedonSchro¨dinger’sformalism 70
5.3 Superpositionstateandwavepacketoscillation 73
5.4 Photons 75
5.5 Problems 84
vii
viii Contents
6 Thehydrogenatom 86
6.1 TheHamiltonianofthehydrogenatom 86
6.2 Angularmomentumofthehydrogenatom 87
6.3 Solutionofthetime-independentSchro¨dingerequationforthe
hydrogenatom 94
6.4 Structureofthehydrogenatom 97
6.5 Electronspinandthetheoryofgeneralizedangularmomentum 101
6.6 Spin–orbitinteractioninthehydrogenatom 106
6.7 Problems 108
7 Multi-electronionsandtheperiodictable 110
7.1 Hamiltonianofthemulti-electronionsandatoms 110
7.2 Solutionsofthetime-independentSchro¨dingerequationformulti-
electronionsandatoms 112
7.3 Theperiodictable 115
7.4 Problems 118
8 Interactionofatomswithelectromagneticradiation 119
8.1 Schro¨dinger’sequationforelectricdipoleinteractionofatomswith
electromagneticradiation 119
8.2 Time-dependentperturbationtheory 120
8.3 Transitionprobabilities 122
8.4 Selectionrulesandthespectraofhydrogenatomsandhydrogen-likeions 126
8.5 Theemissionandabsorptionprocesses 128
8.6 LightAmplificationbyStimulatedEmissionofRadiation(LASER)
andtheEinsteinA-andB-coefficients 130
8.7 Problems 133
9 Simplemolecularorbitalsandcrystallinestructures 135
9.1 Time-independentperturbationtheory 135
9.2 Covalentbondingofdiatomicmolecules 139
9.3 sp,sp2,andsp3orbitalsandexamplesofsimpleorganicmolecules 144
9.4 Diamondandzincblendestructuresandspacelattices 148
9.5 Problems 149
10 Electronicpropertiesofsemiconductorsandthep-njunction 151
10.1 Molecularorbitalpictureofthevalenceandconductionbandsof
semiconductors 151
10.2 Nearly-free-electronmodelofsolidsandtheBlochtheorem 153
10.3 Thek-spaceandtheEvs.kdiagram 157
10.4 Density-of-statesandtheFermienergyforthefree-electrongasmodel 163
10.5 Fermi–Diracdistributionfunctionandthechemicalpotential 164
10.6 Effectivemassofelectronsandholesandgroupvelocityin
semiconductors 170