Modern quantum mechanics

525 Pages·5.964 MB·English

by  Napolitano, Jim J., Sakurai, Sakurai Jun John

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Modern Quantum Mechanics

J.J. Sakurai Jim J. Napolitano

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ISBN 978-1-29202-410-3
9 781292 024103
Modern Quantum Mechanics
J.J. Sakurai Jim J. Napolitano
Second Edition
ISBN 10: 1-292-02410-0
ISBN 13: 978-1-292-02410-3
Pearson Education Limited
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ISBN 10: 1-292-02410-0
ISBN 13: 978-1-292-02410-3
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
123344555656195901119917993357
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Chapter 1. Fundamental Concepts
J. J. Sakurai/Jim J. Napolitano 1
2. Chapter 2. Quantum Dynamics
J. J. Sakurai/Jim J. Napolitano 67
3. Chapter 3. Theory of Angular Momentum
J. J. Sakurai/Jim J. Napolitano 159
4. Chapter 4. Symmetry in Quantum Mechanics
J. J. Sakurai/Jim J. Napolitano 269
5. Chapter 5. Approximation Methods
J. J. Sakurai/Jim J. Napolitano 311
6. Chapter 6. Scattering Theory
J. J. Sakurai/Jim J. Napolitano 397
7. Chapter 7. Identical Particles
J. J. Sakurai/Jim J. Napolitano 459
Appendix A: Electromagnetic Units
J. J. Sakurai/Jim J. Napolitano 499
Appendix B: Brief Summary of Elementary Solutions to Schrödinger’s Wave Equation
J. J. Sakurai/Jim J. Napolitano 503
Bibliography
J. J. Sakurai/Jim J. Napolitano 513
Index 515
I
II
Fundamental Concepts
The revolutionary change in our understanding of microscopic phenomena that
tookplaceduringthe first 27yearsof the twentiethcenturyis unprecedentedin
thehistoryofnaturalsciences.Notonlydidwewitnessseverelimitationsinthe
validityofclassicalphysics,butwefoundthealternativetheorythatreplacedthe
classicalphysicaltheoriestobefarbroaderinscopeandfarricherinitsrangeof
applicability.
Themosttraditionalwaytobeginastudyofquantummechanicsistofollow
thehistoricaldevelopments—Planck’sradiationlaw,theEinstein-Debyetheoryof
specificheats,theBohratom,deBroglie’smatterwaves,andsoforth—together
with careful analyses of some key experimentssuch as the Compton effect, the
Franck-Hertz experiment, and the Davisson-Germer-Thompson experiment. In
thatwaywemaycometoappreciatehowthephysicistsinthefirstquarterofthe
twentiethcenturywereforcedtoabandon,little bylittle, thecherishedconcepts
ofclassicalphysicsandhow,despiteearlierfalsestartsandwrongturns,thegreat
masters—Heisenberg,Schrödinger,andDirac,amongothers—finallysucceeded
informulatingquantummechanicsasweknowittoday.
However, we do not follow the historical approach in this text. Instead, we
start with an examplethatillustrates, perhapsmorethan anyother example,the
inadequacyof classical conceptsin a fundamentalway. We hope that, exposing
readersto a “shocktreatment”at theonsetwillresultin their becomingattuned
towhatwemightcallthe“quantum-mechanicalwayofthinking”ataveryearly
stage.
This different approach is not merely an academic exercise. Our knowledge
ofthephysicalworldcomesfrommakingassumptionsaboutnature,formulating
theseassumptionsintopostulates,derivingpredictionsfromthosepostulates,and
testing such predictions against experiment. If experiment does not agree with
the prediction, then, presumably, the original assumptions were incorrect. Our
approachemphasizesthefundamentalassumptionswemakeaboutnature,upon
whichwe havecometo baseallof ourphysicallaws, andwhichaim toaccom-
modateprofoundlyquantum-mechanicalobservationsattheoutset.
1 THESTERN-GERLACHEXPERIMENT
The examplewe concentrateon in this section is the Stern-Gerlachexperiment,
originallyconceivedby O.Stern in 1921andcarriedoutin Frankfurtbyhimin
FromChapter1ofModernQuantumMechanics,SecondEdition.J.J.Sakurai,JimJ.Napolitano.
Copyright©2011byPearsonEducation,Inc.Allrightsreserved.
1
FundamentalConcepts
What was
actually observed Silver atoms
Classical
prediction
N
S
Furnace
Inhomogeneous
magnetic field
FIGURE1 TheStern-Gerlachexperiment.
∗
collaborationwithW.Gerlachin1922. Thisexperimentillustratesinadramatic
manner the necessity for a radical departure from the concepts of classical me-
chanics.Inthesubsequentsectionsthebasicformalismofquantummechanicsis
presented in a somewhat axiomatic manner but always with the example of the
Stern-Gerlachexperimentinthebackofourminds.Inacertainsense,atwo-state
systemoftheStern-Gerlachtypeistheleastclassical,mostquantum-mechanical
system.Asolidunderstandingofproblemsinvolvingtwo-statesystemswillturn
out to be rewarding to any serious student of quantum mechanics. It is for this
reasonthatwereferrepeatedlytotwo-stateproblemsinthistext.
DescriptionoftheExperiment
WenowpresentabriefdiscussionoftheStern-Gerlachexperiment,whichisdis-
cussed in almost every book on modern physics.† First, silver (Ag) atoms are
heated in an oven.The oven has a small hole throughwhich some of the silver
atomsescape. Asshownin Figure1, thebeam goesthrougha collimatorandis
then subjected to an inhomogeneousmagnetic field produced by a pair of pole
pieces,oneofwhichhasaverysharpedge.
We must now work out the effect of the magnetic field on the silver atoms.
For our purpose the following oversimplified model of the silver atom suffices.
Thesilveratomismadeupofanucleusand47electrons,where46outofthe47
electronscan be visualized as forminga sphericallysymmetricalelectroncloud
withnonetangularmomentum.Ifweignorethenuclearspin,whichisirrelevant
toourdiscussion,weseethattheatomasawholedoeshaveanangularmomen-
tum, which is due solely to the spin—intrinsic as opposed to orbital—angular
∗
ForanexcellenthistoricaldiscussionoftheStern-Gerlachexperiment,see“SternandGerlach:
How a Bad Cigar Helped Reorient Atomic Physics,” by Bretislav Friedrich and Dudley Her-
schbach,PhysicsToday,December(2003)53.
†ForanelementarybutenlighteningdiscussionoftheStern-Gerlachexperiment,seeFrenchand
Taylor(1978),pp.432–38.
2
FundamentalConcepts
momentumofthesingle47th(5s)electron.The47electronsareattachedtothe
nucleus,whichis∼2×105timesheavierthantheelectron;asaresult,theheavy
atom as a whole possesses a magnetic moment equal to the spin magnetic mo-
mentofthe47thelectron.Inotherwords,themagneticmomentμoftheatomis
proportionaltotheelectronspinS,
μ∝S, (1.1)
wherethepreciseproportionalityfactorturnsouttobee/m c(e<0inthistext)
e
toanaccuracyofabout0.2%.
Becausetheinteractionenergyofthemagneticmomentwiththemagneticfield
isjust−μ·B,thez-componentoftheforceexperiencedbytheatomisgivenby
∂ ∂B
F = (μ·B)(cid:6)μ z, (1.2)
z ∂z z ∂z
where we have ignored the components of B in directions other than the z-
direction.Becausetheatomasawholeisveryheavy,weexpectthattheclassical
conceptoftrajectorycanbelegitimatelyapplied,apointthatcanbejustifiedus-
ingtheHeisenberguncertaintyprincipletobederivedlater.Withthearrangement
ofFigure1, the μ >0 (S <0) atomexperiencesa downwardforce,whilethe
z z
μ <0 (S >0) atom experiencesan upwardforce. The beam is then expected
z z
togetsplitaccordingtothevaluesofμ .Inotherwords,theSG(Stern-Gerlach)
z
apparatus“measures”thez-componentofμor,equivalently,thez-componentof
Suptoaproportionalityfactor.
Theatomsintheovenarerandomlyoriented;thereisnopreferreddirectionfor
theorientationofμ.Iftheelectronwerelikeaclassicalspinningobject,wewould
expectallvaluesofμ toberealizedbetween|μ|and−|μ|.Thiswouldleadusto
z
expectacontinuousbundleofbeamscomingoutoftheSGapparatus,asindicated
inFigure1,spreadmoreorlessevenlyovertheexpectedrange.Instead,whatwe
experimentallyobserveis morelike the situationalso shownin Figure1, where
two“spots”areobserved,correspondingtoone“up”andone“down”orientation.
Inotherwords,theSGapparatussplitstheoriginalsilverbeamfromtheoveninto
twodistinctcomponents,aphenomenonreferredtointheearlydaysofquantum
theory as “space quantization.” To the extent that μ can be identified within a
proportionality factor with the electron spin S, only two possible values of the
z-componentofSareobservedtobepossible:S upandS down,whichwecall
z z
S +and S −.Thetwopossiblevaluesof S aremultiplesofsomefundamental
z z z
unit of angular momentum; numerically it turns out that S =h¯/2 and −h¯/2,
z
where
h¯ =1.0546×10−27erg-s
(1.3)
= 6.5822×10−16eV-s.
∗
This“quantization”oftheelectronspinangularmomentum isthefirstimportant
featurewededucefromtheStern-Gerlachexperiment.
∗
Anunderstandingoftherootsofthisquantizationliesintheapplicationofrelativitytoquantum
mechanics.
3
FundamentalConcepts
(a) (b)
FIGURE2 (a)ClassicalphysicspredictionforresultsfromtheStern-Gerlachexper-
iment.Thebeamshouldhavebeenspreadoutvertically,overadistancecorresponding
totherangeofvaluesofthemagneticmomenttimesthecosineoftheorientationangle.
SternandGerlach,however,observedtheresultin(b),namelythatonlytwoorientations
ofthemagneticmomentmanifestedthemselves.Thesetwoorientationsdidnotspanthe
entireexpectedrange.
Figure 2a shows the result one would have expected from the experiment.
Accordingtoclassicalphysics,thebeamshouldhavespreaditselfoveravertical
distance correspondingto the (continuous)range of orientation of the magnetic
moment.Instead,oneobservesFigure2b,whichiscompletelyatoddswithclassi-
calphysics.Thebeammysteriouslysplitsitselfintotwoparts,onecorresponding
tospin“up”andtheothertospin“down.”
Ofcourse,thereisnothingsacredabouttheup-downdirectionorthez-axis.We
couldjustaswellhaveappliedaninhomogeneousfieldinahorizontaldirection,
sayinthex-direction,withthebeamproceedinginthey-direction.Inthismanner
wecouldhaveseparatedthebeamfromtheovenintoan S +componentandan
x
S −component.
x
SequentialStern-GerlachExperiments
Let us now consider a sequential Stern-Gerlach experiment. By this we mean
thattheatomicbeamgoesthroughtwoormoreSGapparatusesinsequence.The
firstarrangementweconsiderisrelativelystraightforward.We subjectthebeam
comingoutoftheoventothearrangementshowninFigure3a,whereSGzˆstands
for an apparatus with the inhomogeneous magnetic field in the z-direction, as
usual.We thenblockthe S − componentcomingoutofthe firstSGzˆ apparatus
z
andlettheremainingS +componentbesubjectedtoanotherSGzˆapparatus.This
z
timethereisonlyonebeamcomponentcomingoutofthesecondapparatus—just
the S +component.Thisisperhapsnotsosurprising;afterall,iftheatomspins
z
areup,theyareexpectedtoremainso,shortofanyexternalfieldthatrotatesthe
spinsbetweenthefirstandthesecondSGzˆ apparatuses.
AlittlemoreinterestingisthearrangementshowninFigure3b.Herethefirst
SG apparatus is the same as before, but the second one (SGxˆ) has an inhomo-
geneousmagnetic field in the x-direction.The S + beam thatenters the second
z
apparatus (SGxˆ) is now split into two components, an S + component and an
x
4
FundamentalConcepts
S+ comp.
z
S+ comp.
z
Oven SGzˆ SGzˆ
No S– comp.
S– comp. z
z
(a)
S+ beam
z S+ beam
x
Oven SGzˆ SGxˆ
S– beam
S– beam x
z
(b)
S+ beam S+ beam
z x
S+ beam
z
Oven SGzˆ SGxˆ SGzˆ
S– beam
z
Sz– beam Sx– beam
(c)
FIGURE3 SequentialStern-Gerlachexperiments.
S −component,with equalintensities. Howcanwe explainthis?Doesitmean
x
that50%of the atomsin the S + beamcomingoutof the first apparatus(SGzˆ)
z
are madeupof atomscharacterizedbyboth S + and S +, while the remaining
z x
50%haveboth S +and S −?Itturnsoutthatsuchapicturerunsintodifficulty,
z x
aswewillseebelow.
We now consider a third step, the arrangement shown in Figure 3c, which
most dramatically illustrates the peculiarities of quantum-mechanical systems.
This time we add to the arrangementof Figure 3b yet a third apparatus, of the
SGzˆ type. It is observed experimentally that two components emerge from the
thirdapparatus,notone;theemergingbeamsareseentohavebothanS +compo-
z
nentandan S −component.Thisisacompletesurprisebecauseaftertheatoms
z
emergedfromthefirstapparatus,wemadesurethattheS −componentwascom-
z
pletely blocked.How is it possible that the S − component,which we thought,
z
weeliminatedearlier,reappears?Themodelinwhichtheatomsenteringthethird
apparatusarevisualizedtohavebothS +andS +isclearlyunsatisfactory.
z x
Thisexampleisoftenusedtoillustratethatinquantummechanicswecannot
determine both S and S simultaneously. More precisely, we can say that the
z x
selection of the S + beam by the second apparatus (SGxˆ) completely destroys
x
anypreviousinformationaboutS .
z
Itisamusingtocomparethissituationwiththatofaspinningtopinclassical
mechanics,wheretheangularmomentum
L=Iω (1.4)
can be measured by determiningthe componentsof the angular-velocityvector
ω.Byobservinghowfasttheobjectisspinninginwhichdirection,wecandeter-
mineω ,ω ,andω simultaneously.ThemomentofinertiaIiscomputableifwe
x y z
5