The Picture Book of Quantum Mechanics

492 Pages·2012·13.46 MB·English

by  Siegmund Brandt

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.