Table Of ContentJaroslav Zamastil · Jakub Benda
Translated with the assistance of Tereza Uhlí˘rová
Quantum
Mechanics and
Electrodynamics
Quantum Mechanics and Electrodynamics
Jaroslav Zamastil • Jakub Benda
TranslatedwiththeassistanceofTerezaUhl´ırˇova´
Quantum Mechanics
and Electrodynamics
123
JaroslavZamastil JakubBenda
CharlesUniversityinPrague CharlesUniversityinPrague
Prague,CzechRepublic Prague,CzechRepublic
ThisisanEnglish-languagetranslationoftheCzech-languageoriginalpublication:Kvantova´
mechanika a elektrodynamika by Jaroslav Zamastil and Jakub Benda; Karolinum Press;
©UniverzitaKarlovavPraze,2016;©JaroslavZamastil,JakubBenda,2016.
ISBN978-3-319-65779-0 ISBN978-3-319-65780-6 (eBook)
DOI10.1007/978-3-319-65780-6
LibraryofCongressControlNumber:2017949677
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Preface
Wehavenotredefinedquantumtheory;wecarryittoitslogicalconclusion.(...)Welearned
itsecondorthirdhand,asanestablisheddisciplinewhoserulesandtechniqueswecameto
feelasintuitiveandnatural,notasapeculiardisplacementofclassical:wefoundandfind
italmostpainfultodo19thcenturyphysics.ThegreatBohr-Einsteinphilosophicaldebates
whichfascinatehistoriansandthephilosophersaretousabitwrong-headed(...) [1]
AFewWordsofExplanation
Nowadays,whenthereisalargenumberofbooksonquantummechanicsavailable,
someofthemareindeedofhighqualityandrightfullyregardedasthe“classics”(e.g.
[4,8]fortheuninitiated,[7,19,24]forintermediateand[15]foradvancedreaders),
whenotherremarkableexpositionsappear(e.g.[25]),whenthereisalargenumber
ofbooksonquantumelectrodynamicsandquantumfieldtheory,someofthemare
indeed excellent (ranging widely from easier pieces such as [3,12,18,23,26] to
moredifficultonessuchas[2,5,13,21,27]),andwhenalmostanytextbook ever
writtencanbefoundontheInternet,downloadedandprinted,wefeelaneedtostart
withanexcuse,oratleastwithafewwordsofexplanation,forcomingupwithyet
anotherbookonquantummechanicsandelectrodynamics.
1. Symmetries of a problem at hand (such as parity, angular momentum etc.)
and their application in the solutions of quantum mechanical problems (e.g.
the Wigner-Eckart theorem) are usually subject to such an abstract exposition
thatanuninitiatedreader,evenifheunderstandstheconcept,doesnotseetheir
usefulness.Forthisreason,wehavedevotedconsiderableattentiontofindingthe
solutions to the simplest, physically interesting problems that cannot be solved
exactly, such as the anharmonic oscillator and the helium atom. In contrast to
other textbooks, we carry out the solutions to their very end and provide more
thananoutlineofthem.Today,whenroutinesforthediagonalizationofmatrices
form a standard part of libraries of advanced programming languages such as
v
vi Preface
Maple,Matlab,Mathematica,Octave,Scilab,Maximaetc.,wewanttoshow
thereaderhowsuchproblemscanbesolvedaloneusingalaptop.
Furthermore,wewanttoemphasizeyetanotheraspectoftheuseofsymmetry
inquantummechanics:namelytheadvantageofanalgebraicoverananalytical
approach to both precisely solvable and insolvable problems. The algebraic
approach provides an elegant solution to the very few physically interesting
problems that can be solved exactly, such as the harmonic oscillator, angular
momentum,compositionofangularmomentaandhydrogen-likeatoms.Itenables
a complete solution to be constructed, including the pertinent wave functions
withoutknowledgeofvariousorthogonalpolynomialsetc.Itsadvantagebecomes
even more obvious when dealing with problems that cannot be solved exactly;
namely, the calculation of the Hamiltonian matrix elements for variational
calculation of anharmonic oscillators, helium and more complex atoms, the
inclusion of the continuous part of the spectra in the case of complex atoms
etc.
It is needless to emphasize that the problem of symmetries and their
applicationsinquantummechanicsisnotaminorone,norisitatallcomplicated.
Once the basic mathematical concepts have been absorbed that are needed
to understand quantum mechanics, such as the concept of an operator, a
commutator, eigenstates and eigenvalues, one is able to understand everything
about symmetries in addition to their applications, without any recourse to the
mathematicalnicetiesofgrouptheory.
2. We wanted to explain relativistic quantum electrodynamics with minimal
necessary formalism and with an emphasis on its physical content and its
applicationstoatomicphysics.
Modernexpositionsofquantumelectrodynamics,suchas[18,21,27],1heavily
rely on either the Dyson-Wick expansion or path integrals, both of which are
undoubtedly very powerful tools. However, for an uninitiated reader, both the
physical content of the formalism and its connection to the usual formalism
of quantum mechanics are rather, if not completely, obscured. Moreover, the
applications are almost exclusively restricted to the high-energy scattering
processes.
Two conclusions could thus be drawn. First, it may appear that the “low-
energy”physicistsdonotneedtopossessanyknowledgeofquantumfieldtheory.
Second,theusualformalismofquantummechanicswouldhavetobedismissed
at the very beginning and a completely new formalism of the quantum field
theory would have to be learnt. However, this task is clearly too difficult and
bearing in mind the first point, it is not worth the effort. Furthermore, when
dealingwiththemostimportantlow-energyquantumelectrodynamicprocesses,
suchasspontaneousemissionandatom-photonscattering,itsufficestoconsider
thenon-relativistictheory,whichismuchmorecomprehensible.
1Wefindthemexcellentandstronglyrecommendthemtothereader.
Preface vii
We do not share this view for a variety of reasons. First and foremost,
even experimenters could know the theoretical value contained in precise
measurements. In addition, once beyond the first approximation, even within
the non-relativistic theory, the so-called virtual processes are encountered.
However,theseprocessescannotbefullydescribedwithinnon-relativistictheory.
Therefore,thistheoryisclearlyincomplete.Moreover,methodsofquantumfield
theoryhaveprovedtobeofenormoususefarbeyondtheapplicationforwhich
they were primarily invented. For instance, they find their applications in both
classicalandquantummany-bodyproblems,seeforexample[16,17,27].Finally,
we believe that quantum field theory—like the general theory of relativity—is
partofthebasiceducationofeveryphysicist.
When dealing with relativistic quantum electrodynamics, we first proceed
within the framework of the ordinary quantum-mechanical formalism and only
then show how the mathematical description is simplified by adopting the
Feynman view of positrons as electrons running backwards in time. Although
thisderivationisneithertheshortestnorthemostformallysatisfying,itenables
one to determine the key steps in the transition from the non-relativistic to the
relativistictheory,toseewhereintheFeynmandiagramsandrulestheordinary
perturbationmethodandCoulomblawarehiddenetc.Theapproachadoptedin
thisbookliesclosesttothatinthebook[23].
We made a considerable effort to minimize the discussion of purely formal
aspectsofproblemsontheonehandandtocompletelyavoidthephrase“itcanbe
shown that” on the other hand. This means that we tried to carefully motivate
and derive everything within a physicist’s level of rigor. This also means that
a number of (important) topics are deliberately treated in a manner that can be
justifiably considered superficial. In particular, this applies to our treatment of the
spinors,adequateformalismofrelativisticquantumfieldtheory,renormalization,Lie
algebrasetc.EventhediscussionofLorentzandgaugeinvariancedoesnotgointo
muchdepth.Therearenecessarilyomissionsineverybook;apartoftheprocessof
writingistodecidewhatshouldbeomittedandwhatshouldbeincluded.Thechoice
ofthethemesandtheirexpositioninourworkwasdictatedbyourstrongconviction
aboutwhatshouldbelearntinthefirstplace.Thepurposeofeveryphysicaltheory
worthy of that name is to “get the numbers out”. Despite the knowledge of deep
connectionshavingitsvalue,webelievethatoneshouldlearnhowthetheoryworks
inthefirstplace,i.e.hownumberscanbeobtainedthatexperimenterscanmeasure.
Regardingthelevelofrigor,wewanttostressthatthemathematicallevelofrigor
iscompletelyalientous.Theartofapproximationformsanintegralpartof“theart
of physics” and we strongly feel that concepts should be explained to others in a
waythatisascloseaspossibletohowwethinkaboutthemourselves.
If the reader is not satisfied either with our choice of the topics or with our
method of exposition, he is encouraged to consult other textbooks, for instance,
thosementionedatthebeginningofthisPreface.Thebooks[15]and[2]areofan
encyclopaedicnature.
viii Preface
Regardingtheliterature,werefertoafewbasicbooksweconsidertobegenerally
knownandavailable.Thereissuchalargenumberofbooksonquantummechanics
that even a complete list of them exceeds our abilities. In the first three chapters,
where we introduce the fundamentals of quantum mechanics, we do not list our
inspirations. However, we try to do so in the advanced parts of this book. Note,
though,thattheabsenceofacitationdoesnotnecessarilyimplythatourexposition
isoriginal.
There are a number of exercises throughout the text. They are intended for the
reader to examine how much he understood. The exercises are, according to the
difficulty, denoted by a number of . If the reader is able to complete only one
exercise,heissurelyabletorecognizewhetheragreaternumberof marksamore
orlessdifficultexercise.
Finally,inthisbookwefocusontheexpositionofthe“classics”withinquantum
mechanics and electrodynamics. Note, though, that the word “classics” does not
implythatthesetopicsceasedtobeofinterest.Forinstance,theNobelprizeof2012
was awarded for experiments that manipulated the individual mesoscopic objects
and thus probed the boundary between the quantum and classical behaviour, see
for example [6]. In 2005, the Nobel prize was awarded for extremely accurate
spectroscopicmeasurements.Whencombinedwiththetheoreticalpredictionsbased
onquantumelectrodynamics,thesemeasurementsenableustodeducethenuclear
properties, such as the proton size, with substantially better accuracy than by any
other means. This in turn leads to surprising findings, see for example [22]. Thus,
whatfollowsarethe“classics”;nevertheless,theclassicsarestillalive!2
Prerequisites
Thefollowingismainlyforautodidacts,whohaveoursympathies.Shouldthisbook
bringjoyandnotfrustration,thefollowingshouldhold.
1. The reader has a good command of single-variable calculus, complex numbers
andbasiclinearalgebra.
2. The reader possesses some knowledge of multi-variable calculus and vector
analysis. The level of exposition in for example, Feynman’s freshman lectures
[9]shouldsuffice.
3. For parts concerning quantum electrodynamics, namely Chapters 6 and 7, it is
necessarytoknowthecomplexanalysisatthelevelofforexample [10].
2Wedonotgoasfarastotrytoexplainthelatestdevelopments,suchasthetwomentionedabove.
Webelievethatbymasteringthecontentofthisbook,thereaderwillbeabletocatchuponthem
onhisown.
Preface ix
4. Thereadershouldpossesssomeknowledgeofclassicalmechanics,electrodynam-
icsandspecialrelativity.Again,knowledgeatthelevelof[9]shouldsuffice.Some
knowledgeoftheHamiltonformulationofclassicalmechanicswouldbeuseful,
thoughnotcrucial.The“classic”textbooksare[11,14].
Acknowledgments
Wewouldliketoexpressourgratitude
(cid:129) ToJakubZahumensky´,LadaVybulkova´ andToma´sˇJavu˚rekfortheirhelpwhen
creatingtheveryfirstversionofthemanuscript.
(cid:129) To Roman Cˇur´ık and Toma´sˇ Mancˇal for a number of suggestions and
improvements.
(cid:129) Toourdearmentor,Prof.Lubom´ırSka´la,forhissupportofthewholeprojectin
itsinitialphase.Unfortunately,hepassedawaybeforecompletion.
Errors
Anup-to-datelistoferrorswehavenotdiscoveredintimewillbekeptat
http://quantum.karlov.mff.cuni.cz/(cid:2)jzamastil/
Prague,CzechRepublic JaroslavZamastil
JakubBenda
Contents
1 FoundationsofQuantumMechanics ...................................... 1
1.1 BasicPrinciples ......................................................... 1
1.2 MathematicalSchemeoftheQuantumTheory........................ 5
1.2.1 Stern-GerlachExperiments ................................... 5
1.2.2 Operators....................................................... 13
1.2.3 TimeEvolutioninQuantumTheory.......................... 14
1.2.4 StationaryStates............................................... 15
1.2.5 PropertiesofHermitianOperators............................ 17
1.2.6 AmbiguityintheDeterminationofStates.................... 20
1.2.7 RabiMethodofMagneticMoments.......................... 21
1.3 SystemswithMoreDegreesofFreedom .............................. 23
1.3.1 ExpectedValuesofOperatorsandTheirTimeEvolution... 23
1.3.2 CanonicalQuantization ....................................... 25
1.3.3 HarmonicOscillator........................................... 27
1.3.4 AbstractSolution.............................................. 29
1.3.5 MatrixRepresentation......................................... 31
1.3.6 Diracı-Function............................................... 33
1.3.7 CoordinateRepresentation.................................... 34
1.3.8 MomentumRepresentation ................................... 37
1.3.9 GaussianPacketandtheUncertaintyPrinciple.............. 39
1.4 FinalNotes .............................................................. 42
References..................................................................... 42
2 ApproximateMethodsinQuantumMechanics .......................... 45
2.1 VariationalMethod...................................................... 46
2.1.1 TheRitzVariationalPrinciple ................................ 46
2.1.2 OptimizationofNonlinearParameters....................... 47
2.1.3 OptimizationofLinearParameters........................... 48
xi
