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This page intentionally left blank 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 cb2 2ru The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521829526 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 isbn-13 978-0-511-12595-9 eBook (NetLibrary) isbn-10 0-511-12595-x eBook (NetLibrary) isbn-13 978-0-521-82952-6 hardback isbn-10 0-521-82952-6 hardback url Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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