Quantum Physics: The Bottom-Up Approach: From the Simple Two-Level System to Irreducible Representations

281 Pages·2013·3.54 MB·English

by  Dirk Dubbers

Graduate Texts in Physics
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Dirk Dubbers (cid:2) Hans-Jürgen Stöckmann
Quantum Physics:
The Bottom-Up
Approach
From the Simple Two-Level System
to Irreducible Representations
DirkDubbers Hans-JürgenStöckmann
Fak.PhysikundAstronomie FachbereichPhysik
PhysikalischesInstitut Philipps-UniversitätMarburg
UniversitätHeidelberg Marburg,Germany
Heidelberg,Germany
ISSN1868-4513 ISSN1868-4521(electronic)
GraduateTextsinPhysics
ISBN978-3-642-31059-1 ISBN978-3-642-31060-7(eBook)
DOI10.1007/978-3-642-31060-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012950772
©Springer-VerlagBerlinHeidelberg2013
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Preface
Quantummechanicspervadesmanybranchesofscience,fromphysics,materialsci-
ence,informatics,tochemistryandmolecularbiology.Manyproductsofeveryday
lifederivefromdiscoveriesbasedonquantumphysics,likesiliconchips,magnetic
storagedevices,lasers,medicalimagingdevices,aswellaschemicalsandbiochem-
icals.Therefore,scientistsandengineersinmanyfieldsneedagoodunderstanding
of quantum theory, but often they are overwhelmed by the sheer volume of most
standardtextbooksonquantumphysics.
Our approach is to first limit discussion to the smallest systems in nature that
still display the basic features of quantum theory. Hence this tutorial at first deals
with systems in which only two quantum states are involved, subject to external
perturbations. Such effective spin one-half systems are a valuable training ground
to elucidate the subtleties of quantum theory and, indeed, the essence of quantum
mechanicsliesinthetwo-levelsystem.Wepresentbasicquantumcalculationsstep
bystepinasimplenotationandinsufficientdetailbecausethepractitionerusually
hasnotimetolosewhenproceedingfromoneequationtothenext.
As a starting point we assume the reader to have taken introductory courses in
quantum mechanics and linear algebra, and to be familiar with the Schrödinger
equationandtheessentialsofangularmomentum,whichwerecapitulateinPartIof
thebook.PartIIcoversessentialtopicsofquantumphysicsbasedonthetwo-state
approach, including subjects of high contemporary interest as quantum entangle-
ment,quantumchaos,orgeometricphases.InPartIII,theresultsthenareappliedto
varioustopicsfromatomic,condensedmatter,andnuclearphysics,andfromquan-
tuminformatics.
We then proceed to the more general concepts of quantum theory. To this end,
PartIVofthistreatiserestartsfromfirstprinciplestodevelopthetheoryofangular
momentum,sphericaltensors,andirreduciblerepresentations.Wederiveageneral-
izedspinprecessionequationthatcoversthehighermultipoleinteractions,andap-
plytheresultstovarioustopicsinatomicandcondensedmatterphysics.Chapterson
multiplequantumtransitions,dressedatomeffects,spinrelaxationanddecoherence
concludethetutorial.
v
vi Preface
Thetextisbasedonvariouslecturesgivenbytheauthorsonthetwo-statesystem,
irreducibletensors,andquantumchaos,andiscomplementedwithanillustrativeset
ofbasicexperiments,manyofthemdoneintheauthors’respectivelaboratories.In
short,theaimofthistutorialistoprovidethebachelorstudentaswellasthepracti-
tionerwithacompacttextthatletsthemunderstandawealthofquantumphysics.
Heidelberg,Germany DirkDubbers
Marburg,Germany Hans-JürgenStöckmann
“DürermeetsEinstein,travelling.”InstallationbySabrinaHohmann,foraninterpre-
tationseehttp://www.physi.uni-heidelberg.de/KunstamBau.pdf.Withfriendlyper-
missionoftheartist.Foto:Wolf-DieterGericke
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Contents
PartI Prologue
1 RecollectionsfromElementaryQuantumPhysics . . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
PartII Two-StateQuantumSystems
2 AMostSimpleTwo-LevelSystem . . . . . . . . . . . . . . . . . . . . 11
2.1 MagneticMomentandSpin . . . . . . . . . . . . . . . . . . . . . 11
2.2 ZeemanEffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Stern–GerlachEffect . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 QuantumTheoryinaNutshell . . . . . . . . . . . . . . . . . . . . . 17
3.1 SpinMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 EnergyMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 ExpectationValues . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 ExpectationValueofEnergy . . . . . . . . . . . . . . . . 22
3.3.2 ExpectationValueofSpin . . . . . . . . . . . . . . . . . . 23
3.4 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.1 UncertaintyofSpin . . . . . . . . . . . . . . . . . . . . . 26
3.4.2 UncertaintyofEnergy . . . . . . . . . . . . . . . . . . . . 27
3.5 SpinPrecession . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.1 LongitudinalField . . . . . . . . . . . . . . . . . . . . . . 28
3.5.2 TransverseField . . . . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 ExperimentsonSpinPrecession . . . . . . . . . . . . . . . . . . . . . 33
4.1 MuonSpinPrecession . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 LightScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 SpinorRotationThrough720°. . . . . . . . . . . . . . . . . . . . 38
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
ix