The Picture Book of Quantum Mechanics PDF
Preview The Picture Book of Quantum Mechanics
The Picture Book of Quantum Mechanics Siegmund Brandt • Hans Dieter Dahmen The Picture Book of Quantum Mechanics Fourth Edition Siegmund Brandt Hans Dieter Dahmen Physics Department Physics Department University of Siegen University of Siegen D-57068 Siegen D-57068 Siegen Germany Germany [email protected] [email protected] Please note that additional material for this book can be downloaded from http://extras.springer.com ISBN 978-1-4614-3950-9 ISBN 978-1-4614-3951-6 (eBook) DOI 10.1007/978-1-4614-3951-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012937850 © Springer Science+Business Media New York 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface to the Fourth Edition InthepresenteditiontheillustrationsofthePictureBookappearinfullcolor. The scope of the book was extended again. There is now a chapter on hy- bridizationandsectionsonboundstatesandonscatteringinpiecewiselinear potentialsinonedimension. Fromthewebpagewww.extras.springer.comallillustrationscan bedownloadedforeasyuseinlecturesandseminars.TheinclusionofaCD- ROMwithwiththatmaterial(asinthethirdedition)isnolongernecessary. TogeneratethecomputergraphicsofthefirsteditionofthePictureBook, we developed an interactive program on quantum mechanics. A modernized version, which we call INTERQUANTA (abbreviated IQ), together with an accompanying text has been published by Springer in various editions. The most recent one1 can be regarded as companion to this Picture Book. It al- lowsinteractivemanipulationofofahostofphysicsandgraphicsparameters and produces output in the form of static and moving pictures. The program runs under Windows, Linux and Mac OS X. It is a pleasure to acknowledge the generous help provided by IBM Germany in the development of IQ. In particular, we want to thank Dr. U. Groh for his competent help in the early phaseofthework. All computer-drawn figures in the present edition were produced using the published version of IQ or extensions realized with the help of Mr. Anli Shundi,Dr.SergeiBoris,andDr.TiloStroh. Siegen,January2012 SiegmundBrandt HansDieterDahmen 1S.Brandt,H.D.Dahmen,andT.StrohInteractiveQuantumMechanics–QuantumEx- perimentsontheComputer,2nded.,Springer,NewYork,2011 v Foreword Students of classical mechanics can rely on a wealth of experience from ev- eryday life to help them understand and apply mechanical concepts. Even though a stone is not a mass point, the experience of throwing stones cer- tainlyhelpsthemtounderstandandanalyzethetrajectoryofamasspointina gravitational field. Moreover, students can solve many mechanical problems on the basis of Newton’s laws and, in doing so, gain additional experience. When studying wave optics, they find that their knowledge of water waves, aswellasexperimentsinarippletank,isveryhelpfulinforminganintuition aboutthetypicalwavephenomenaofinterferenceanddiffraction. Inquantummechanics,however,beginnersarewithoutanyintuition.Be- cause quantum-mechanical phenomena happen on an atomic or a subatomic scale,wehavenoexperienceofthemindailylife.Theexperimentsinatomic physicsinvolvemoreorlesscomplicatedapparatusandarebynomeanssim- ple to interpret. Even if students are able to take Schrödinger’s equation for granted, as many students do Newton’s laws, it is not easy for them to ac- quire experience in quantum mechanics through the solution of problems. Onlyveryfewproblemscanbetreatedwithoutacomputer.Moreover,when solutionsinclosedformareknown,theircomplicatedstructureandthespecial mathematicalfunctions,whichstudentsareusuallyencounteringforthefirst time,constitutesevereobstaclestodevelopingaheuristiccomprehension.The most difficult hurdle, however, is the formulation of a problem in quantum- mechanicallanguage,fortheconceptsarecompletelydifferentfromthoseof classicalmechanics.Infact,theconceptsandequationsofquantummechan- ics in Schrödinger’s formulation are much closer to those of optics than to thoseofmechanics.Moreover,thequantitiesthatweareinterestedin–such astransitionprobabilities,crosssections,andsoon–usuallyhavenothingto do with mechanical concepts such as the position, momentum, or trajectory of a particle. Nevertheless, actual insight into a process is a prerequisite for understanding its quantum-mechanical description and for interpreting basic propertiesinquantummechanicslikeposition,linearandangularmomentum, aswellascrosssections,lifetimes,andsoon. vii viii Foreword Actually, students must develop an intuition of how the concepts of clas- sical mechanics are altered and supplemented by the arguments of optics in order to acquire a roughly correct picture of quantum mechanics. In partic- ular, the time evolution of microscopic physical systems has to be studied to establish how it corresponds to classical mechanics. Here, computers and computer graphics offer incredible help, for they produce a large number of examplesthatareverydetailedandthatcanbelookedatinanyphaseoftheir time development. For instance, the study of wave packets in motion, which is practically impossible without the help of a computer, reveals the limited validityofintuitiondrawnfromclassicalmechanicsandgivesusinsightinto phenomena like the tunnel effect and resonances, which, because of the im- portance of interference, can be understood only through optical analogies. Avarietyofsystemsindifferentsituationscanbesimulatedonthecomputer andmadeaccessiblebydifferenttypesofcomputergraphics. Someofthetopicscoveredare • scatteringofwavepacketsandstationarywavesinonedimension, • thetunneleffect, • decayofmetastablestates, • boundstatesinvariouspotentials, • energybands, • distinguishableandindistinguishableparticles, • angularmomentum, • three-dimensionalscattering, • crosssectionsandscatteringamplitudes, • eigenstates in three-dimensional potentials, for example, in the hydro- genatom,partialwaves,andresonances, • motionofwavepacketsinthreedimensions, • spinandmagneticresonance. Conceptualtoolsthatbridgethegapbetweenclassicalandquantumconcepts include • thephase-spaceprobabilitydensityofstatisticalmechanics, • theWignerphase-spacedistribution, • theabsolutesquareoftheanalyzingamplitudeasprobabilityorproba- bilitydensity. Foreword ix Thegraphicalaidscomprise • timeevolutionsofwavefunctionsforone-dimensionalproblems, • parameterdependencesforstudying,forexample,thescatteringovera rangeofenergies, • three-dimensional surface plots for presenting two-particle wave func- tionsorfunctionsoftwovariables, • polar(antenna)diagramsintwoandthreedimensions, • plotsofcontourlinesorcontoursurfaces,thatis,constantfunctionval- ues,intwoandthreedimensions, • ripple-tankpicturestoillustratethree-dimensionalscattering. Whenever possible, how particles of a system would behave according to classical mechanics has been indicated by their positions or trajectories. In passing,thespecialfunctionstypicalforquantummechanics,suchasLegen- dre, Hermite, and Laguerre polynomials, spherical harmonics, and spherical Besselfunctions,arealsoshowninsetsofpictures. Thetextpresentstheprincipalideasofwavemechanics.Theintroductory Chapter 1 lays the groundwork by discussing the particle aspect of light, us- ingthefundamentalexperimentalfindingsofthephotoelectricandCompton effects and the wave aspect of particles as it is demonstrated by the diffrac- tionofelectrons.Thetheoreticalideasabstractedfromtheseexperimentsare introducedinChapter2bystudyingthebehaviorofwavepacketsoflightas they propagate through space and as they are reflected or refracted by glass plates. Chapter 3 introduces material particles as wave packets of de Broglie waves. The ability of de Broglie waves to describe the mechanics of a parti- cleisexplainedthroughadetaileddiscussionofgroupvelocity,Heisenberg’s uncertaintyprinciple,andBorn’sprobabilityinterpretation.TheSchrödinger equationisfoundtobetheequationofmotion. Chapters 4 through 9 are devoted to the quantum-mechanical systems in one dimension. Study of the scattering of a particle by a potential helps us understandhowitmovesundertheinfluenceofaforceandhowtheprobabil- ityinterpretationoperatestoexplainthesimultaneouseffectsoftransmission andreflection.Westudythetunneleffectofaparticleandtheexcitationand decayofametastablestate.Acarefultransitiontoastationaryboundstateis carried out. Quasi-classical motion of wave packets confined to the potential rangeisalsoexamined. Thevelocityofaparticleexperiencingthetunneleffecthasbeenasubject of controversial discussion in the literature. In Chapter 7, we introduce the concepts of quantile position and quantile velocity with which this problem canbetreated.
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