Table Of ContentGraduate Texts in Physics
Graduate Texts in Physics
Graduate Texts in Physics publishes core learning/teaching material for graduate- and
advanced-level undergraduate courses on topics of current and emerging fields within
physics,bothpureandapplied.ThesetextbooksservestudentsattheMS-orPhD-leveland
theirinstructorsascomprehensivesourcesofprinciples,definitions,derivations,experi-
mentsandapplications(asrelevant)fortheirmasteryandteaching,respectively.Interna-
tionalinscopeandrelevance,thetextbookscorrespondtocoursesyllabisufficientlytoserve
asrequiredreading.Theirdidacticstyle,comprehensivenessandcoverageoffundamental
materialalsomakethemsuitableasintroductionsorreferencesforscientistsentering,or
requiringtimelyknowledgeof,aresearchfield.
SeriesEditors
ProfessorWilliamT.Rhodes
FloridaAtlanticUniversity
DepartmentofComputerandElectricalEngineeringandComputerScience
(cid:127)
ImagingScienceandTechnologyCenter
777GladesRoadSE,Room456
BocaRaton,FL33431,USA
E-mail:[email protected]
ProfessorH.EugeneStanley
BostonUniversity
CenterforPolymerStudies
DepartmentofPhysics
590CommonwealthAvenue,Room204B
Boston,MA02215,USA
E-mail:[email protected]
ProfessorRichardNeeds
CavendishLaboratory
JJThomsonAvenue
CambridgeCB30HE,UK
E-mail:[email protected]
PleaseviewavailabletitlesinGraduateTextsinPhysicsonserieshomepage
http://www.springer.com/series/8431/
Daniel R. Bes
Quantum Mechanics
A Modern and Concise Introductory Course
ThirdEdition
With 63 Figures
123
Daniel R.Bes
ComisionNacionaldeEnergiaAtomica
Av.GeneralPaz1499
1650Prov. de Buenos Aires, Argentina
Universidad Favaloro
Solis 453
1078 C.A.B.A., Argentina
ISSN1868-4513 e-ISSN1868-4521
ISBN978-3-642-20555-2 e-ISBN978-3-642-20556-9
DOI10.1007/978-3-642-20556-9
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2012938371
©Springer-VerlagBerlinHeidelberg2004, 2007, 2012
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationor
partsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,inits
currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer .Violationsareliable to
prosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
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SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Aprimevalrepresentationofthehydrogenatom
This beautiful mandala is displayed at the temple court of Paro Dzong, the
monumental fortress of Western Bhutan [1]. It may be a primeval representation
of the hydrogen atom: the outer red circle conveys a meaning of strength, which
maycorrespondtothe electronbindingenergy.Theinnerandsphericalnucleusis
surroundedbylarge,osculatingcirclesthatrepresentthemotionoftheelectron:the
circlesnotonlyoccupyafiniteregionofspace(asinFig.6.4)butarealsoassociated
with trajectories of different energies (colors) and/or with radiation transitions of
different colors (wavelengths). At the center, within the nucleus, there are three
quarks.Realitycandisplaycharmingcoincidenceswithreligious/artisticcreations.
Seealsofigureattheendofthechapters.
v
(cid:127)
Foreword
Quantummechanicsisundergoingarevolution.Notthatitssubstanceischanging,
but major developments are placing it in the focus of renewed attention, both
withinthephysicscommunityandamongthescientificallyinterestedpublic.First,
wonderfully clever table-top experiments involving the manipulation of single
photons,atomic particles and molecules are revealingin an ever-moreconvincing
manner theoretically predicted facts about the counterintuitive and sometimes
“spooky” behavior of quantum systems and have led to a renewed interest in the
formulationofastrictlyphysics-based(non-philosophical)foundationofquantum
mechanics.Second,theprospectofbuildingquantumcomputerswith enormously
increasedcapacityof information-processingis developinggreatinterestand high
hopes in the engineering and computer science communities. Third, condensed
matterphysicsandnanotechnologyare narrowingdownthe gapbetweenclassical
and quantum domains in the practical realm. These developments demand more
and better training in quantum mechanics at the universities, with emphasis on a
clear and solid understandingof the subject. Cookbook-stylelearning of quantum
mechanics,inwhichequationsandmethodsfortheirsolutionarememorizedrather
thanunderstood,may helpthe studentsto solve somestandardproblemsand pass
multiple-choice tests, but it will not enable them to break new ground in real
life as physicists. On the other hand, some “Mickey Mouse courses” on quantum
mechanicsforengineers,biologistsandcomputeranalystsmaygiveanideaofwhat
thisdisciplineisabout,buttoooftenthestudentendsupwithanincorrectpictureor,
at best, a bunch of uncritical,blind beliefsin linear algebrawizardry.The present
book represents a fresh start toward helping achieve a deep understanding of the
subject. It presents the material with utmost rigor and will require ironclad, old-
fashioneddisciplinefromthestudentsintheirstudy.
Too frequently, in today’s universities, we hear the demand that the courses
offeredbe“entertaining,”inresponsetowhichsomedepartmentalbrochuresdeclare
that “physics is fun”! Studying physics requires many hours of hard work, deep
concentration, long discussions with buddies, periodic consultation with faculty,
and tough self-discipline. But it can, and should, become a passion: the passion
toachievea deepunderstandingofhowNatureworks.Thisunderstandingusually
vii
viii Foreword
comes in discrete steps, and students will experience such a step-wise mode of
progressastheyworktheirwaydiligentlythroughthepresentbook.Thesatisfaction
ofhavingsuccessfullymasteredeachstepwillthenindeedfeelveryrewarding!
The“amountofinformationperunitsurface”oftextisveryhighinthisbook–
its pagescoverall the importantaspectsof present-dayquantummechanics,from
theone-dimensionalharmonicoscillatortoteleportation.Startingfromafewbasic
principles and concentrating on the fundamental behavior of quantum systems
(particles)withonlyafewdegreesoffreedom(low-dimensionHilbertspace)allows
the author to plunge right into the core of quantum mechanics. It also makes it
possible to introduce first the Heisenberg matrix approach – in my opinion, a
pedagogically rewarding method that helps sharpen the mental and mathematical
tools needed in this discipline right at the beginning. For instance, solving the
quantization of the harmonic oscillator without the recourse of a differential
equation is illuminating and teaches dexterity in handling the vector and matrix
representationofstatesandoperators,respectively.
DanielBesisachildoftheCopenhagenschool.Honedinoneofthecradlesof
quantummechanicsby A˚ge Bohr,son of the greatmaster, andby Ben Mottelson,
hedevelopedanunusuallyacuteunderstandingofthesubject,whichafteryearsof
maturinghenowhasprojectedintoabook–nowinitsthirdedition.Theemphasis
giventhroughoutthetexttothefundamentalroleandmeaningofthemeasurement
process will help the student overcome the initial reaction to the counterintuitive
aspects of quantum mechanics and better comprehend the physical meaning and
properties of Schro¨dinger’s wave function. The human brain is an eminently
classicalsystem(albeitthemostcomplexoneintheUniverseasweknowit),whose
phylo-andontogeneticevolutionaredrivenbyclassicalinteractionsbetweenorgan-
ism and the environment, and which processes pragmatic information, definable
onlyintheclassicaldomain.Itisthereforeonlynaturalthatwhenthisclassicalbrain
looksintothemicroscopicdomain,usinghuman-designedinstrumentswhichmust
translate quantum happenings into classical, microscopically observable effects,
strangethingswithunfamiliarbehaviormayappearinourmentalimages!Anditis
onlynaturalthat,thus,theobserver’swayofthinkingandinstrumentsshouldnotbe
leftcompletelyoutsidethefundamentalframeworkofquantumphysics!Bes’book
helpstorecognize,understandandacceptquantum“paradoxes”notassuchbutas
thefactsof“Natureunderhumanobservation.”Oncethisacceptancehassettledin
themind,thestudentwillhavedevelopeda trueintuitionor,astheauthorlikesto
callit,a“feeling”forquantummechanics.
Chapter 2 contains the real foundation on which quantum mechanics is built;
it thus deserves, in my opinion, repeated readings – not just at the beginning
but after each subsequent chapter. With the exception of the discussion of two
additionalprinciples,therestofthebookdescribesthemathematicalformulations
of quantum mechanics (both the Heisenberg matrix mechanics, most suitable for
the treatment of low-dimension state vectors, and Schro¨dinger’s wave mechanics
for continuous variables) as well as many up-to-date applications. The examples
cover a wide variety of topics, from the simple harmonic oscillator mentioned
abovetosubjectsincondensedmatterphysics,nuclearphysics,electrodynamicsand
Foreword ix
quantumcomputing.Itisinterestingtonote,regardingthelatter,thattheconceptof
qubitappearsin a most naturalway in the middle of the book, almost in passing,
well before the essence of quantum computing is discussed towards the end. Of
particularhelparethecarefullythought-outproblemsattheendofeachchapter,as
wellastheoccasionallistingsof“commonmisconceptions.”Amostwelcometouch
istheinclusionofafinalchapteronthehistoryoftheoreticalquantummechanics–
indeed, it is regrettable that so little attention is given to it in university physics
curricula: much additional understanding can be gained from learning how ideas
havematured(orfailed)duringthehistoricaldevelopmentofagivendiscipline!
Let me conclude with a personal note. I have known Daniel Bes for almost
70 years. As a matter of fact, I had known of him even before we met in person:
ourfatherswere“commuter–trainacquaintances”inBuenosAires,andbothserved
on the PTA of the primaryschoolthatDaniel and I attended(in differentgrades).
Daniel and I were physics students at the University of Buenos Aires (again,
at different levels), then on the physics faculty, and years later, visiting staff
membersofLosAlamosNationalLaboratory.Wealwayswerefriends,butwenever
worked together – Daniel was a theoretician almost from the beginning, whereas
I started as a cosmic-ray and elementary-particle experimentalist (see Fig.2.7).
Interestingly, reading the first edition of this book has motivated me a few years
ago to expand my current interest in foundationsof information theory and work
towardsdevelopingyetanotherinterpretationof quantummechanics(whichI call
“information-based”)!
Itgivesmeaparticularpleasurethataftersomanydecadesanddespiteresiding
at opposite ends of the American continent, we have become professionally
“entangled”throughthiswonderfultextbook!
Fairbanks JuanG.Roederer
