Theoretical Physics 6. Quantum Mechanics - Basics

526 Pages·2017·5.927 MB·English

by  Wolfgang Nolting

Wolfgang Nolting
Theoretical
Physics 6
Quantum Mechanics - Basics
Theoretical Physics 6
Wolfgang Nolting
Theoretical Physics 6
Quantum Mechanics - Basics
123
WolfgangNolting
Inst.Physik
Humboldt-UniversitätzuBerlin
Berlin,Germany
ISBN978-3-319-54385-7 ISBN978-3-319-54386-4 (eBook)
DOI10.1007/978-3-319-54386-4
LibraryofCongressControlNumber:2016943655
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General Preface
TheninevolumesoftheseriesBasicCourse:TheoreticalPhysicsarethoughttobe
textbookmaterialforthestudyofuniversitylevelphysics.Theyareaimedtoimpart,
in a compact form, the most important skills of theoretical physics which can be
usedasbasisforhandlingmoresophisticatedtopicsandproblemsintheadvanced
study of physics as well as in the subsequent physics research. The conceptual
designofthepresentationisorganizedinsuchawaythat
ClassicalMechanics(volume1)
AnalyticalMechanics(volume2)
Electrodynamics(volume3)
SpecialTheoryofRelativity(volume4)
Thermodynamics(volume5)
are considered as the theory part of an integrated course of experimental and
theoretical physics as is being offered at many universities starting from the first
semester.Therefore,thepresentationisconsciouslychosentobeveryelaborateand
self-contained,sometimessurelyatthe costofcertainelegance,sothatthecourse
is suitableevenforself-study,at firstwithoutanyneedof secondaryliterature.At
anystage,nomaterialisusedwhichhasnotbeendealtwithearlierinthetext.This
holds in particular for the mathematical tools, which have been comprehensively
developed starting from the school level, of course more or less in the form of
recipes,suchthatrightfromthebeginningofthestudy,onecansolveproblemsin
theoreticalphysics. The mathematicalinsertionsare always then pluggedin when
theybecomeindispensabletoproceedfurtherintheprogramoftheoreticalphysics.
It goes without saying that in such a context, not all the mathematical statements
canbeprovedandderivedwithabsoluterigor.Instead,sometimesareferencemust
be made to an appropriate course in mathematics or to an advanced textbook in
mathematics. Nevertheless, I have tried for a reasonably balanced representation
so that the mathematical tools are not only applicable but also appear at least
“plausible”.
v
vi GeneralPreface
Themathematicalinterludesareofcoursenecessaryonlyinthefirstvolumesof
thisseries,whichincorporatemoreorlessthematerialofabachelorprogram.Inthe
secondpartoftheserieswhichcomprisesthemodernaspectsoftheoreticalphysics,
QuantumMechanics:Basics(volume6)
QuantumMechanics:MethodsandApplications(volume7)
StatisticalPhysics(volume8)
Many-BodyTheory(volume9),
mathematical insertions are no longer necessary. This is partly because, by the
time one comes to this stage, the obligatory mathematics courses one has to take
in order to study physics would have provided the required tools. The fact that
training in theory has already started in the first semester itself permits inclusion
of parts of quantum mechanics and statistical physics in the bachelor program
itself. It is clear that the content of the last three volumes cannot be part of an
integrated course but rather the subject matter of pure theory lectures. This holds
in particular for Many-Body Theory which is offered, sometimes under different
names,e.g.,AdvancedQuantumMechanics,in the eighthorso semester ofstudy.
In this part, new methods and concepts beyond basic studies are introduced and
discussed which are developed in particular for correlated many particle systems
whichinthemeantimehavebecomeindispensableforastudentpursuingamaster’s
orahigherdegreeandforbeingabletoreadcurrentresearchliterature.
In all the volumes of the series Theoretical Physics, numerous exercises are
included to deepen the understanding and to help correctly apply the abstractly
acquired knowledge. It is obligatory for a student to attempt on his own to adapt
and apply the abstract concepts of theoretical physics to solve realistic problems.
Detailedsolutionstotheexercisesaregivenattheendofeachvolume.Theideais
tohelpastudenttoovercomeanydifficultyataparticularstepofthesolutionorto
checkone’sowneffort.Importantlythesesolutionsshouldnotseducethestudentto
followtheeasywayoutasasubstituteforhisowneffort.Attheendofeachbigger
chapter,Ihaveaddedself-examinationquestionswhichshallserveasaself-testand
maybeusefulwhilepreparingforexaminations.
I should not forget to thank all the people who have contributed one way or
another to the success of the book series. The single volumes arose mainly from
lectures which I gave at the universities of Muenster, Wuerzburg, Osnabrueck,
and Berlin (Germany),Valladolid (Spain), and Warangal(India). The interest and
constructive criticism of the students provided me the decisive motivation for
preparing the rather extensive manuscripts. After the publication of the German
version,Ireceivedalotofsuggestionsfromnumerouscolleaguesforimprovement,
and this helped to further develop and enhance the concept and the performance
of the series. In particular, I appreciate very much the support by Prof. Dr. A.
Ramakanth,a long-standingscientific partnerand friend,who helpedme in many
respects,e.g.,whatconcernsthecheckingofthetranslationoftheGermantextinto
thepresentEnglishversion.
GeneralPreface vii
SpecialthanksareduetotheSpringercompany,inparticulartoDr.Th.Schneider
and his team. I remember many useful motivations and stimulations. I have the
feelingthatmybooksarewelltakencareof.
Berlin,Germany WolfgangNolting
August2016
Preface to Volume 6
Themaingoalofthepresentvolume6(QuantumMechanics:Basics)corresponds
exactly to that of the total basic course in Theoretical Physics. It is thought to
be accompanyingtextbook material for the study of university-levelphysics. It is
aimedtoimpart,inacompactform,themostimportantskillsoftheoreticalphysics
which can be used as basis for handling more sophisticated topics and problems
in the advanced study of physics as well as in the subsequent physics research.
It is presented in such a way that it enables self-study without the need for a
demanding and laborious reference to secondary literature. For the understanding
of the text it is only presumed that the reader has a good grasp of what has been
elaboratedintheprecedingvolumes.Mathematicalinterludesarealwayspresented
in a compact and functional form and practiced when they appear indispensable
forthe furtherdevelopmentof the theory.Forthe wholetextit holdsthatI had to
focusontheessentials,presentingtheminadetailedandelaborateform,sometimes
consciouslysacrificingcertainelegance.Itgoeswithoutsaying,thatafterthebasic
course, secondaryliterature is needed to deepenthe understandingof physicsand
mathematics.
For the treatment of Quantum Mechanics also, we have to introduce certain
new mathematicalconcepts. However now, the special demands may be of rather
conceptual nature. The Quantum Mechanics utilizes novel ‘models of thinking’,
which are alien to Classical Physics, and whose understandingand applying may
raisedifficultiestothe‘beginner’.Therefore,inthiscase,itisespeciallymandatory
tousetheexercises,whichplayanindispensableroleforaneffectivelearningand
thereforeareofferedafterallimportantsubsections,inordertobecomefamiliarwith
theatfirstunaccustomedprinciplesandconceptsoftheQuantumMechanics.The
elaboratesolutionsto exercisesat the end ofthe bookshould notkeep the learner
fromattemptinganindependenttreatmentoftheproblems,butshouldonlyserveas
acheckupofone’sownefforts.
This volume on QuantumMechanicsarose from lecturesI gaveat the German
universitiesinWürzburg,Münster,andBerlin.Theanimatinginterestofthestudents
inmylecturenoteshasinducedmetopreparethetextwithspecialcare.Thepresent
ix
x PrefacetoVolume6
oneaswellastheothervolumesisthoughttobethetextbookmaterialforthestudy
ofbasicphysics,primarilyintendedforthestudentsratherthanfortheteachers.
Thewealthofsubjectmatterhasmadeitnecessarytodividethepresentationof
Quantum Mechanics into two volumes, where the first part deals predominantly
with the basics. In a rather extended first chapter, an inductive reasoning for
Quantum Mechanics is presented, starting with a critical inspection of the ‘pre-
quantum-mechanicaltime’,i.e.,withananalysisoftheproblemsencounteredbythe
physicists at the beginningof the twentieth century.Surely,opinionson the value
ofsuchahistoricalintroductionmaydiffer.However,Ithinkitleadstoaprofound
understandingofQuantumMechanics.
ThepresentationandinterpretationoftheSchrödingerequation,thefundamental
equationofmotionofQuantumMechanics,whichreplacestheclassicalequations
of motion (Newton, Lagrange, Hamilton), will be the central topic of the second
chapter. The Schrödinger equation cannot be derived in a mathematically strict
sense, but has rather to be introduced, more or less, by analogy considerations.
For this purposeone can, for instance,use the Hamilton-Jacobitheory (section 3,
Vol. 2), according to which the Quantum Mechanics should be considered as
something like a super-ordinate theory, where the Classical Mechanics plays a
similarroleintheframeworkofQuantumMechanicsasthegeometricalopticsplays
inthegeneraltheoryoflightwaves.Theparticle-wavedualismofmatter,oneofthe
mostdecisivescientificfindingsofphysicsinthetwentiethcentury,willalreadybe
indicatedviasuchan‘extrapolation’ofClassicalMechanics.
The second chapter will reveal why the state of a system can be described by
a ‘wave function’, the statistical character of which is closely related to typical
quantum-mechanical phenomena as the Heisenberg uncertainty principle. This
statisticalcharacterofQuantumMechanics,incontrasttoClassicalPhysics,allows
for only probability statements. Typical determinants are therefore probability
distributions,averagevalues,andfluctuations.
The Schrödinger wave mechanics is only one of the several possibilities to
represent Quantum Mechanics. The complete abstract basics will be worked out
inthethirdchapter.While inthefirstchaptertheQuantumMechanicsisreasoned
inductively,whicheventuallyleadstotheSchrödingerversioninthesecondchapter,
now,opposite,namely,thedeductivewaywillbefollowed.Fundamentaltermssuch
as state and observable are introduced axiomatically as elements and operators
of an abstract Hilbert space. ‘Measuring’ means ‘operation’ on the ‘state’ of the
system, as a result of which, in general, the state is changed. This explains why
the describing mathematics represents an operator theory, which at this stage of
the course has to be introduced and exercised. The third chapter concludes with
someconsiderationsonthecorrespondenceprinciplebywhichoncemoretiesare
establishedtoClassicalPhysics.
In the fourth chapter, we will interrupt our general considerations in order to
deepen the understanding of the abstract theory by some relevant applications to
simplepotentialproblems.Asimmediateresultsofthemodelcalculations,wewill
encountersomenovel,typicalquantum-mechanicalphenomena.Therewiththefirst
part of the introductionto Quantum Mechanics will end. Further applications, in-
PrefacetoVolume6 xi
depthstudies,andextensionsofthesubjectmatterwillthenbeofferedinthesecond
part:TheoreticalPhysics7:QuantumMechanics—MethodsandApplications.
I am thankful to the Springer company, especially to Dr. Th. Schneider, for
acceptingandsupportingtheconceptofmyproposal.Thecollaborationwasalways
delightfulandveryprofessional.Adecisivecontributiontothebookwasprovided
byProf.Dr.A.RamakanthfromtheKakatiyaUniversityofWarangal(India).Many
thanksforit!
Berlin,Germany WolfgangNolting
November2016