Solved problems in quantum and statistical mechanics

409 Pages·2012·2.646 MB·English

by  Michele Cini, Francesco Fucito, Mauro Sbragaglia (auth.)

SolvedProblemsinQuantumandStatisticalMechanics

Michele Cini
Francesco Fucito
Mauro Sbragaglia
Solved Problems
in Quantum
and Statistical Mechanics
MicheleCini FrancescoFucito MauroSbragaglia
DepartmentofPhysics DepartmentofPhysics DepartmentofPhysics
UniversityofRome UniversityofRome UniversityofRome
TorVergata, TorVergataandINFN TorVergataandINFN
INFN
LaboratoriNazionaliFrascati
UNITEXT-CollanadiFisicaeAstronomia
ISSNprintedition:2038-5730 ISSNelectronicedition:2038-5765
ISBN978-88-470-2314-7 e-ISBN978-88-470-2315-4
DOI10.1007/978-88-470-2315-4
LibraryofCongressControlNumber:2011940537
SpringerMilanDordrechtHeidelbergLondonNewYork
(cid:2)c Springer-VerlagItalia2012
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Preface
ItalianstudentsstartstudyingQuantumandStatisticalMechanicsinthelastyearof
theirundergraduatestudies.Manyphysiciststhinkthesesubjectsarethecoreofan
educationinphysics.Atthesametime,thesetwosubjectsarenoteasilylearntby
theaveragestudent.InItalythefinalexamisdividedintwoseparateparts:thereis,
infact,awrittenandanoralone.Mosttextbooksconcentrateontheprinciplesofthe
theoryandtheapplicationsaredealtwithattheendofeachchapterunderthehead-
ings‘problems’or‘exercises’.Mostofthetimesthelatterconsistonlyofthetextof
theproblemwithsomevagueindicationsonhowtoproceedwiththesolution.Some
othertimesthesolutioniscompletelylefttothestudent.Theauthorsofthepresent
book think this is didactically wrong: these applications are crucial for a correct
understanding of the subject and help the students get acquainted with the mathe-
matical tools they have learnt in other classes. Many times we have noticed that a
simplechangeinthedenominationsoftheletterswasenoughtothrowastudentin
disarray: a function whose behaviour was familiar when the independent variable
wascalledx,becameunfathomablewhentheverysamevariablewastheenergyE.
Inreality,duringtheelementarystudyofclassicalphysics,theexercisesaremostly
straightforwardapplicationsofthegeneralformulaededucedfromexperienceorat
the most require the simplest notions of differential and integral calculus. Things
changewithQuantumandStatisticalMechanicswhosemathematicalformalismis
more complex. Also the problems and exercises reflect this point and the required
solutionsoftenneedlongerandmoreelaboratemanipulations.
It is for these reasons that in our teaching we have always dedicated a large
amountoftimetothediscussionoftheapplications,tothecorrectionoftheprob-
lems,andhavetriedtoelaboratewrittensolutionswithlengthydiscussionstohelp
the students get ready for the written exams. From this point to the publication of
ournotesithasbeenanaturalstep.
vi Preface
Acknowledgements
FrancescoFucitowishestospeciallythankM.Guagnelliwithwhomhehasshared
theteachingofStatisticalMechanicsattheUniversita`diRomaTorVergataformany
years and with whom he started this project which, for various reasons, could not
beendedtogether.HealsowantstothankM.G.DeDivitiisforcollaborationinan
early stage of this work. Francesco Fucito and Mauro Sbragaglia are also grateful
to their students in the classes of Statistical Mechanics in the years 2007−2010
and in particular we wish to thank Cristina Bertulli, Andrea Bussone, Roberta De
Angelis,GiulioDeMagistris,DavideLiberati,SebastianGrothans,ValerioLatini,
FrancescaMancini,StefanoMarchesani,ClaudiaNarcisi,FrancescoNazzaro,Clau-
diaViolante.Theirquestionsandsupporthavebeenofvaluablehelpforus.
Rome,September2011 MicheleCini
FrancescoFucito
MauroSbragaglia
Contents
PartI TheoreticalBackground
1 SummaryofQuantumandStatisticalMechanics .................. 3
1.1 OneDimensionalSchro¨dingerEquation........................ 3
1.2 OneDimensionalHarmonicOscillator......................... 5
1.3 VariationalMethod ......................................... 6
1.4 AngularMomentum ........................................ 8
1.5 Spin...................................................... 10
1.6 HydrogenAtom............................................ 10
1.7 SolutionsoftheThreeDimensionalSchro¨dingerEquation ........ 12
1.8 WKBMethod ............................................. 19
1.9 PerturbationTheory ........................................ 21
1.10 ThermodynamicPotentials................................... 23
1.11 FundamentalsofEnsembleTheory............................ 26
1.11.1 MicrocanonicalEnsemble............................. 28
1.11.2 CanonicalEnsemble.................................. 28
1.11.3 GrandCanonicalEnsemble............................ 29
1.11.4 QuantumStatisticalMechanics......................... 29
1.12 KineticApproach .......................................... 30
1.13 Fluctuations ............................................... 31
1.14 MathematicalFormulae ..................................... 32
References..................................................... 36
PartII QuantumMechanics–Problems
2 FormalismofQuantumMechanicsandOneDimensionalProblems.. 39
3 AngularMomentumandSpin ...................................113
viii Contents
4 CentralForceField.............................................145
5 PerturbationTheoryandWKBMethod ..........................163
PartIII StatisticalMechanics–Problems
6 ThermodynamicsandMicrocanonicalEnsemble ..................193
7 CanonicalEnsemble............................................227
8 GrandCanonicalEnsemble .....................................289
9 KineticPhysics ................................................301
10 Bose-EinsteinGases ............................................315
11 Fermi-DiracGases .............................................337
12 FluctuationsandComplements ..................................363
Index .............................................................393
Part I
Theoretical Background