量子力学英文课件格里菲斯Chapter4

Outline

Thegeneralizationtothreedimensionsisstraight-forward.Schrödinger’sequationsayswheretheHamiltonianoperatorHisobtainedfromtheclassicalenergybythestandardprescription(appliednowtoyandz,aswellasx):orforshort.Thus,Schrödinger’sequation

[4.1]becomesisthe

Laplacian,inCartesiancoordinates.

Thepotentialenergy

Vandthewavefunction

arenowfunctionsofr=(x,y,z)andt.

Theprobabilityoffindingtheparticleininfinitesimalvolume

d3r=dxdydzis|(r,t)|2d3r.withtheintegraltakenoverallspace.

Ifthepotentialisindependentoftime,therewillbeacompletesetofstationarystates,Thenormalizationconditionofthewavefunctionreads

wherethespatialwavefunctionnsatisfiesthetime-independent

Schrödingerequation;

Thegeneralsolutiontothe(time-dependent)

Schrödingerequationiswiththeconstantcndeterminedbytheinitialwavefunction,(r,0),intheusualway.

Ifthepotentialadmitscontinuumstate,thenthesuminEq.[4.9]becomesanintegral.

Typically,thepotentialisafunctiononlyofthedistancefromtheorigin,i.e.V(r,,

)=V(r)

.Inthatcaseitisnaturaltoadoptsphericalcoordinates,(r,,)(seeFigure4.1).

InsphericalcoordinatestheLaplaciantakestheform

Insphericalcoordinates,then,thetime-independent

Schrödingerequationreads

Webeginbylookingforsolutionsthatareseparableintoproducts:PuttingthisintoEq.[4.14],wehaveDividingby(YR)andmultiplyingby-2mr2/ħ2

Theterminthefirstcurlybracketdependsonlyonr,whereastheremainder

dependsonlyon

and;accordingly,eachmustbeaconstant.Forreasonsthatwillappearinduecourse,wewillwritethis“separationconstant”intheforml(l+1).

NOTE:Eqs.[4.16]and[4.17]areequal

tothetime-independent

Schrödingerequation[4.14]!

Eq.[4.17]determinesthedependenceofonand;multiplyingby(Ysin2),itbecomes

TosolveEq.[4.18],wetryseparationofvariables:

PluggingthisinEq.[4.18],anddividingby,wefindThefirsttermisafunctiononlyof,andthesecondisafunctiononlyof,soeach

mustbeaconstant.Thistimewe’llcalltheseparationconstant

m2

:

(i).Theequation[4.21]iseasy:

Now,whenadvancesby2,wereturntothesamepointinspace(seeFigure4.1),soitisnaturaltorequirethatInotherwords,exp[im(+2)]=exp[im],orexp(i2m)=1.Fromthisitfollowsthatm

mustbean

integer

:(ii).Theequation[4.20]

maynotbesofamiliar.Thesolutionis

wherePlmistheassociatedLegendrefunction,definedbyandPl

(x)isthel-th

Legendrepolynomial.ForourpresentpurposesitismoreconvenienttodefinethembytheRodirgues(罗德里格)formula:

Forexample,ThefirstfewLegendrepolynomialswerelistedinTable3.1.Asthenamesuggests,Pl(x)isapolynomial(ofdegreel)inx,andisevenoroddaccordingtotheparityofl.

Moreover,if|m|>l,thenEq.[4.27]saysPlm

=0.Foranygivenl,then,thereare(2l+1)possiblevaluesofm:

NoticethatlmustbeanonnegativeintegerfortheRodriguesformulatomakeanysense.

Butwait!Eq.[4.25]isasecond-orderdifferentialequation:Itshouldhavetwolinearlyindependent

solutions,foranyoldvaluesoflandm.Wherearealltheothersolutions?

Answer:Theyexist,ofcourse,asmathematicalsolutionstotheequation,buttheyarephysically

unacceptablebecausetheyblowupat=0and/or=,anddonotyieldnormalizable

wavefunctions

(seeProblem4.4).

Now,thevolumeelementinsphericalcoordinatesissothenormalizationcondition(Eq.[4.6])becomesItisconvenienttonormalizeRandYindividually:

Thenormalizedangularwavefunctionsarecalledsphericalharmonics:where

=(-1)mform≥0and

=1

form≤0.Asweshallproveon,theyareautomaticallyorthogonal,soIntable4.2thereislistedthefirstfewsphericalharmonics.Noticethattheangularpartofthewavefunction,Y(,),isthesameforallsphericallysymmetricpotentials.

Theactualshapeofthepotential,V(r),affectsonlytheradialpartofthewavefunction,

R(r),whichisdeterminedbyEq.[4.16]:Thisequationsimplifiesifwechangevariables:Letthus,Eq.[3.35]becomesThisiscalledtheradialequation;itisidenticalinformtotheone-dimensionalSchrödingerequation(Eq.[2.4]),exceptthattheeffectivepotential,containsanextrapiece,centrifugalterm,(ħ2/2m)[l(l+1)/r2].

Ittendstothrowtheparticleoutward(awayfromtheorigin),justlikethecentrifugal(pseudo-)forceinclassicalmechanics.Meanwhile,thenormalizationcondition(Eq.[4.31])becomesWecannotproceedfurtheruntilaspecificpotentialisprovided.Example.Considerthe

infinitesphericalwellOutsidethewellthewavefunctioniszero;insidethewelltheradialequationsays

where

Ourproblemistosolvethisequation,subjecttotheboundarycondition:

u(a)=0.

Thecasel=0iseasy:Butremember,theactualradialwavefunctionisR(r)=u(r)/r,and[cos(kr)]/rblowsupasr0.SowemustchooseB=0.Theboundaryconditionthenrequiressin(ka)=0,andhenceka=n

forsomeintegern.TheallowedenergiesareevidentlyfromEq.[4.42]

thesameasfortheone-dimensionalinfinitesquarewell(Eq.[2.23]).Normalizingu(r)yields

Noticethatthestationarystatesarelabeledbythree

quantumnumbers,n,l,andm:nlm(r,,).

The

energy,however,dependsonlyonnandl:Enl.

ThegeneralsolutiontoEq.[4.41](foranarbitraryintegerl)isnotsofamiliar:wherejl(x)isthesphericalBesselfunctionoforderl,andnl(x)isthe

sphericalNeumannfunctionoforderl.ThesphericalBesselfunctionandthe

sphericalNeumannfunctionaredefinedasfollows:Forexample,ThefirstfewsphericalBesselandNeumannfunctionsarelistedinTable4.3.

Noticethatforsmallx,since

ThepointisthattheBesselfunctionsarefiniteattheorigin,buttheNeumannfunctionsblowupattheorigin.Accordingly,wemusthaveBl

=0,andhenceThereremainstheboundarycondition,R(a)=0.Evidentlykmustbechosensuchthatthatis,(ka)isazeroofthelth

–ordersphericalBesselfunctions.

NowtheBesselfunctionsareoscillatory(seeFigure4.2);eachonehasaninfinitenumberofzeros.Figure4.2:GraphsofthefirstfoursphericalBesselfunctions.Atanyrate,theboundaryconditionrequiresthatwherenlisthenth

zeroofthe

lth

sphericalBesselfunction.

The

allowed

energies,then,aregivenbyandthewavefunctionsarewiththeconstantAnltobedeterminedbynormalization.

Eachenergylevelis(2l+1)-folddegenerate,sincethereare(2l+1)differentvaluesofmforeachvalueofl(seeEq.[4.29]).Homework:

Problem4.1,Problem4.9.Thehydrogenatomconsistsofaheavy,essentiallymotionlessprotonofcharge+e,togetherwithamuchlighterelectron(charge–e)thatcirclesaroundit,heldinorbitbythemutualattractionofoppositecharges(seeFigure4.3).

FromCoulomb'slaw,thepotentialenergy(inSIunits)is

andtheradialequation(Eq.[4.37])saysOurproblemistosolvethisequationforu(r)anddeterminetheallowedelectronenergies

E.Thehydrogenatomissuchanimportantcasethatwe'llworkthemoutindetailbythemethodweusedintheanalyticalsolutiontotheharmonicoscillator.Incidentally,theCoulombpotential(Eq.[4.52])admitscontinuumstates(withE>0),describingelectron-protonscattering,aswellasdiscreteboundstates,representingthehydrogenatom,butweshallconfineourattentiontothelatter.Ourfirsttaskistotidyupthenotation.LetForboundstates,E<0,so

isreal.DividingEq.[4.53]byE,wehaveThissuggeststhatweletNextweexaminetheasymptoticformofthesolutions.

(1)As

theconstantterminthebracketsdominates,so(approximately)(2)As0

thecentrifugaltermdominates;approximatelythen,

ThenextstepistopeelofftheasymptoticbehaviorEqs.[4.58]and[4.59],introducingthenewfunctionv():Thefirstindicationsarenotauspicious:Then,theradialequation(Eq.[4.56])readsFinally,weassumethesolution,v(),canbeexpressedasapowerseriesin:

Ourproblemistodeterminethecoefficients(a0,al,a2,…).DifferentiatingEq.[4.62]termbytermDifferentiatingagain,InsertingtheseintoEq.[4.61]，wehaveEquatingthecoefficientsoflikepowersyieldsThisrecursionformuladeterminesthecoefficients,andhencethefunction

v():

Westartwitha0=A,andEq.[4.63]givesusa1,puttingthisbackin,weobtaina2,andsoon.Nowlet’sseewhatthecoefficientslooklikeforlarge

j(thiscorrespondstolarge

,wherethehigherpowersdominate).

Inthisregimetherecursionformulasays

soSupposeforamomentthatthisweretheexactresult.Thenandhencewhichblowsupatlarge.Thepositiveexponentialispreciselytheasymptoticbehaviorwedidn’twantinEq.[4.57].

Thereisonlyonewayoutofthisdilemma:Theseriesmustterminate.Theremustoccursomemaximalinteger,jmax,suchthatandbeyondwhichallcoefficientsvanishautomatically.

Evidently(Eq.[4.63])theso-called

principalquantumnumber,wehaveBut0

determinesE(seeEqs.[4.54]and[4.55]):sotheallowedenergiesareThisisthefamous

Bohrformula

-byanymeasurethemostimportantresultinallofquantummechanics.istheso-calledBohrradius.Itfollows(again,fromEq.[4.55])thatCombiningEqs.[4.55]and[4.68],wefindthat

Evidentlythespatial

wavefunctionsforhydrogenarelabeledbythree

quantumnumbers(n,l,andm):where(referringbacktoEqs.[4.36]and[4.60])andv()isapolynomialofdegree,jmax=nl1in,whosecoefficientsaredeterminedbytherecursionformula

Thegroundstate(thatis,thestateoflowestenergy)isthecasen=1;puttingintheacceptedvaluesforthephysicalconstants,wegetEvidentlythebindingenergyofhydrogenis13.6eV.

Example1:

n=1

Equation[4.67]forcesl=0,whencealsom=0(seeEq.[4.29]),soTherecursionformulatruncatesafterthefirstterm(Eq.[4.76]withj=0yieldsa1=0),sov()isaconstant(a0)

andExample2:

n=2Ifn=2,theenergyisthisisthefirstexcitedstate.

Sincewecanhaveeitherl=0(inwhichcasem=0)orl=1(withm=1,0,or+1),sothereareactuallyfourdifferentstatesthatsharethisenergy.Ifl=0,therecursionrelation(Eq.[4.76])gives

sov()=a0(1),andhenceIfl=1,therecursionformulaterminatestheseriesafterasingleterm,sov()isaconstant,andwefindIneachcasetheconstant

a0istobedeterminedbynormalization(seeProblem4.11).

Forarbitraryn，thepossiblevaluesofl(consistentwithEq.[4.67])areForeachl,thereare(2l+1)possiblevaluesofmEq.[4.29],sothetotaldegeneracyoftheenergylevelEnisThepolynomial

v()isafunctionwellknowntoappliedmathematicians;apartfromnormalization,itcanbewrittenasisanassociatedLaguerrepolynomial,andistheqth

Laguerrepolynomial.ThefirstfewLaguerrepolynomialsarelistedinTable4.4.

SomeassociatedLaguerrepolynomialsaregiveninTable4.5.Table4.6Thefirstfewradialwavefunctionsforhydrogen,Rnl(r)Figure4.4GraphsofthefirstfewhydrogenradialwavefunctionsThenormalizedhydrogenwavefunctionsareTheyarenotpretty,butdon’tcomplain-thisisoneoftheveryfewrealisticsystemsthatcanbesolvedatall,inexactclosedform.Aswewillprovelateron,theyaremutuallyorthogonal:

Visualizingthehydrogenwavefunctionsisnoteasy.Chemistsliketodraw“densityplots”,inwhichthebrightnessofthecloudisproportionalto|nlm|2(Fig.4.5).Morequantitativearesurfacesofconstantprobabilitydensity(Fig.4.6).Homework:

Problem4.14,Problem4.15.

Inprinciple,ifyouputahydrogenatomintosomestationarystate

nlm，itshouldstaythereforever.

However,ifyoutickleitslightly,thentheatommayundergoatransitiontosomeotherstationarystate-eitherbyabsorbing

energyandmovinguptoahigher-energystate,orbygivingoff

energyandmovingdown.Theresultisthatacontainerofhydrogengivesofflight(photons),whoseenergycorrespondstothedifferenceinenergybetweentheinitialandfinalstates:

Inpracticesuchperturbationsarealwayspresent;transitions(or,quantumjumps)areconstantlyoccurring.Now,accordingtothe

Planckformula,theenergyofaphotonisproportionaltoitsfrequency

:Meanwhile,thewavelengthisgiven=c/

,soRisknownastheRydbergconstant,andEq.[4.93]istheRydbergformulaforthespectrumofhydrogen.

Itwasdiscoveredempiricallyinthenineteenthcentury,andthegreatesttriumphofBohr’stheorywasitsabilitytoaccountforthisresult-andtocalculateRintermsofthefundamentalconstantsofnature.Transitionstothegroundstate(nf

=1)lieintheultraviolet;theyareknowntospectroscopistsastheLymanseries.

Transitionstothefirstexcitedstate(nf=2)fallinthevisibleregion;theyconstitutetheBaimerseries.Transitionstonf

=3(the

Paschenseries)areintheinfrared,andsoon(seeFigure4.5).

Inclassicalmechanics,theangularmomentumofaparticle(withrespecttotheorigin)isgivenbytheformulawhichistosay,Thecorrespondingquantum

operatorsareobtainedbythestandardprescription(Eq.[4.2]):Inthefollowingsectionswewilldeducetheeigenvaluesandeigenfunctionsoftheseoperators.

LxandL

ydonotcommute;infact[providingatestfunction,f(x,y,z)，forthemtoactupon]:Allthetermscancelinpairs(byvirtueoftheequalityofcross-derivatives)excepttwo:andweconclude(droppingthetestfunction)

Bycyclicpermutationoftheindicesitfollowsalsothat

Fromthesefundamentalcommutationrelationstheentiretheoryofangularmomentumcanbededuced!

EvidentlyLx，Ly,andLzareincompatibleobservables.Accordingtothegeneralizeduncertaintyprinciple(Eq.[3.139]),

or

Itwouldthereforebefutiletolookforstatesthataresimultaneously

eigenfunctionsofLxandofLy.

Ontheotherhand,thesquareofthetotal

angularmomentum,does

commutewithLx:Itfollows,ofcourse,L2alsocommuteswithLyandLz

,

or,morecompactly,

SoL2iscompatiblewitheachcomponentofL,andwecanhopetofindsimultaneous

eigenstatesofL2and(say)Lz:

Next,we’llusea“ladderoperator”techniquetofindtheeigenstatesandeigenvalues,verysimilartotheoneweappliedtotheharmonicoscillatorbackinSection2.3.1.LetItscommutatorwithLzis

Tofindthe

eigenvaluesofangularmomentum

followingIffisaneigenfunctionofL2andLz

,soalsoisL±f.

Proof:ByusingofEq.[4.104],forEq.[4.107]sayssoL±fisaneigenfunctionofL2,withthesameeigenvalue

.

andEq.[4.106]sayssoL±fisaneigenfunctionofLzwiththenew

eigenvalue

ħ.QED

L+iscalledthe“raising”operatorbecauseitincreasestheeigenvalueofLzbyħ.Liscalledthe“lowering”operatorbecauseitlowerstheeigenvaluebyħ.Foragivenvalueof,then,weobtaina“ladder”ofstates,witheach“rung”separatedfromitsneighborsbyoneunitofħintheeigenvalueof

Lz

(seeFigure4.6).

Toascendtheladderweapplytheraisingoperator,andtodescend,theloweringoperator.

Butthisprocesscannotgoonforever:Eventuallywe’regoingtoreachastateforwhichthez-componentexceedsthetotal,andthatcannotbe(seeProblem4.18).

Sotheremustexista“toprung”,ft

,suchthat

LetħlbetheeigenvalueofLz

atthistoprungNowor,puttingittheotherwayaround,Itfollowsthat

ThistellsustheeigenvalueofL2intermsofthemaximum

eigenvalueofLz.

Meanwhile,thereisalso(forthesamereason)abottomrung,fb

,suchthatUsingEquation4.112,wehaveComparingEqs.[4.113]and[4.116],weseethat

EvidentlytheeigenvaluesofLzaremħ

，wheremgoesfromlto+lin

Nintegersteps.Inparticular,itfollowsthatl=

l+N,andhence

Youmightreasonablyguessthatthehalf-integersolutionsarespurious,butitturnsoutthattheyareofprofoundimportance,asweshallseeinthefollowingsections.solmustbe

an

integeror

a

half-integer!

Now,theeigenfunctionsarecharacterizedbythenumberslandm:whereForagivenvalueofl,thereare2l+1differentvaluesofm(i.e.,2l+1“rungs”onthe“ladder”).

FirstofallweneedtorewriteLx，Ly,andLzinsphericalcoordinates.Nowandthegradient,insphericalcoordinates,isButasshownbyFig.4.1:andhenceThusEvidently,Weshallalsoneedtheraisingandloweringoperators:Wearenowinapositiontodeterminetheeigenfunction

—flm(,).Assumeit’saneigenfunctionofLz

,witheigenvalue

ħm:

Hereg()isaconstantofintegration,asfarasisconcerned,butitcanstilldependon.

Adifferentialequarionforg()canbewritten(seethetextbook)inamorefamiliarform(seeSection4.1.2):

whichispreciselytheequationforthe

-dependentpart,(),ofYlm(,)(compareEq.[4.25]).Meanwhile,the-dependentpartoffisidenticalto()(Eq.[4.22]).

Ontheotherhand,fisalsoaneigenfunctionofL2,witheigenvalue

ħ2l(l+1):Conclusion:The

sphericalharmonics

arepreciselythe

(normalized)

eigenfunctionsofL2

and

Lz

.

WhenwesolvedtheSchrödingerequationbyseparationofvariables,inSection4.1,wewereinadvertantlyconstructingsimultaneous

eigenfunctionsofthethree

commutingoperators

H,L2,andLz

:Homework:

Problem4.19,

Problem4.22.

Inclassicalmechanics,arigidobjectadmitstwokindsofangularmomentum:orbital(L=rp),associatedwiththemotionofthecenterofmass,andspin(S=I),associatedwithmotionaboutthecenterofmass.

Forexample,theearthhas

orbitalangularmomentumattributabletoitsannualrevolutionaroundthesun,and

spinangularmomentumcomingfromitsdailyrotationaboutthenorth-southaxis.Inquantummechanics,however,andherethedistinctionisabsolutelyfundamental.

Forexample,inthecaseofhydrogen:

orbitalangularmomentum

associatedwiththemotionoftheelectronaroundthenucleus(anddescribedbythesphericalharmonicsfunctions).

Theelectronisastructurelesspointparticle,anditsspinangularmomentumcannotbedecomposedintoorbitalangularmomentaofconstituentparts.

Sufficeittosaythatelementaryparticlescarryintrinsic

angularmomentum(S)inadditiontotheir“extrinsic”angularmomentum(L)!

Thealgebraictheoryofspinisacarboncopyofthetheoryoforbitalangularmomentum.

Beginningwiththefundamentalcommutationrelations:

Itfollows(asbefore)thattheeigenvectorsofS2andSzsatisfywhereS

Sx

iSy.

Butthistimetheeigenvectorsarenot

sphericalharmonics(they’renotfunctionsofandatall).Thereisnoapriorireasontoexcludethehalf-integervaluesofsandm:

Itsohappensthateveryelementaryparticlehasaspecific

andimmutablevalueofs,whichwecallthespinofthatparticularspecies:pimesons()havespins=0electrons(e)havespins=l/2photons()havespins=ldeltashavespins=3/2gravitons(引力子)havespins=2andsoon.

Bycontrast,theorbitalangularmomentum

quantumnumber

lcantakeonany(integer)valueyouplease,andwillchangefromonetoanotherwhenthesystemisperturbed.

Butspinangularmomentum

quantumnumber

s

isfixed,foranygivenparticle,andthismakesthetheoryofspincomparativelysimple.

Themostimportantcaseiss=1/2,forthisisthespinoftheparticlesthatmakeupordinarymatter(protons,neutrons,andelectrons),aswellasallquarksandallleptons.

Moreover,onceyouunderstandspinl/2,itisasimplemattertoworkouttheformalismforanyhigherspin.Therearejust

two

eigenstates:

Usingtheseasbasisvectors,thegeneralstateofaspin-l/2particlecanbeexpressedasatwo-elementcolumnmatrix(orspinor):

Meanwhile,thespinoperators2×2matrices,whichwecanworkoutbynotingtheireffecton+and:

Eq.[4.135]saysandEq.[4.136]givesNow,S

Sx

iSy,soanditfollowsthat

ThuswhileIt'salittletidiertodivideoffthefactorofħ/2:

ThesearethefamousPaulispinmatrices.NoticethatSx

,

Sy

,

Sz

,andS2

areallHermitian(astheyshouldbe,sincetheyrepresentobservables).

Ontheotherhand,S+

andSare

not

Hermitianevidentlytheyarenotobservable.TheeigenspinorsofSzare(ofcourse)

IfyoumeasureSzonaparticleinthegeneralstate(Eq.[4.139]),youcouldget+ħ/2,withprobability|a|2,orħ/2,withprobability|b|2.

Sincethesearetheonlypossibilities,

Butwhatifyouchosetomeasure

Sx?Whatarethepossibleresults,andwhataretheirrespectiveprobabilities?Accordingtothegeneralizedstatisticalinterpretation,weneedtoknowtheeigenvaluesandeigenspinorsof

Sx.ThecharacteristicequationofSxisNotsurprisingly,thepossiblevaluesfor

Sxarethesameasthosefor

Sz.

Theeigenspinorsareobtainedintheusualway:so,

=

.

Evidentlythe(normalized)eigenspinors

ofSx

areAstheeigenvectorsofaHermitianmatrix,theyspanthespace:thegenericspinor

canbeexpressedasalinearcombinationofthem:Ifyoumeasure

Sxinthestate

:

theprobabilityofgetting+ħ/2is(1/2)|a+b|2,and

theprobabilityofgettingħ/2is(1/2)|ab|2.Example.Supposeaspinl/2particleisinthestateEvidentlytheexpectationvalueofSxis

whichwecouldalsohaveobtainedmoredirectly:Homework:

Problem4.27,Problem4.30.

Aspinningchargedparticleconstitutesamagneticdipole.

Itsmagneticdipolemoment

isproportionaltoitsspinangularmomentum

S:

theproportionalityconstantiscalledthegyromagneticratio(旋磁比).

WhenamagneticdipoleisplacedinamagneticfieldB,itexperiencesatorque(扭转力/转矩),B,whichtendstolineitupparalleltothefield(justlikeacompassneedle).

Theenergyassociatedwiththistorqueis

sotheHamiltonianofaspinningchargedparticle,atrestinamagneticfield

B,becomes

whereSistheappropriatespinmatrix(Eq.[4.147],inthecaseofspin1/2).Example:

Larmorprecession(进动/旋进).

Imagineaparticleofspin1/2atrestinauniformmagneticfield,whichpointsinthez-direction:TheHamiltonian

matrixisTheeigenstatesofHarethesameasthoseofSzEvidently,theenergyislowestwhenthedipolemomentisparalleltothefield一justasitwouldbeclassically.SincetheHamiltonianistimeindependent,thegeneralsolutiontothetime-dependentSchrodingerequation,canbeexpressedintermsofthestationarystates:Theconstantsaandbaredeterminedbytheinitialconditions,say

where|a|2+|b|2=1.

Withnoessentiallossofgenerality,we’llwritewherefixedanglewhosephysicalsignificancewillappearinamoment.

ThusTogetafeelforwhatishappeninghere,let’scalculatetheexpectationvalueofthespinSasafunctionoftime:Similarity,EvidentlySistitledataconstantangletothez-axis,andprecessesaboutthefieldattheLarmorfrequency

justasitwouldclassically(seeFigure4.7).Example:theStern-Gerlachexperiment.

Inaninhomogeneous

magneticfield,thereisnotonlyatorque,butalsoaforce,onamagneticdipole:Thisforcecanbeusedtoseparateoutparticleswithaparticularspinorientation(方向).asfollows.

Imagineabeamofrelativelyheavyneutral(中性的)atoms,travelinginthey-direction,whichpassesthrougharegionofinhomogeneousmagneticfield(Figure4.8)Forexample,whereB0

isastronguniformfieldandtheconstantdescribesasmalldeviationfromhomogeneity.

TheforceontheseatomsisButbecauseoftheLarmorprecessionaboutB0,Sxoscillatesrapidly,andaveragestozero;thenet

forceisinthez-direction:andthebeamisdeflectedupordown,inproportiontothez-componentofthespinangularmomentum.Classicallywe’dexpectasmear,butinfactthebeamsplitsinto2s+1individualbeams,beautifullydemons