Chapter1(1)

Chapter1
Thebasicsof quantummechanics
1.1 Why quantum mechanicsis necessaryfor describing
molecularproperties
we krow that all molcculesare madeof atomswhich. in turn. containnu-
clei andelectrons.As I discussin this introcjuctorysection,the equationsthat
governthe motionsof electronsand of nuclei are not the familiar Newton
equatrons.
F : m a ( l . l )
but a nervsetof equationscalledSchrodingerequations.when scientistsfirst
studiedthe behaviorof electronsand nuclei.thev tried to interprettherrex-
perimentalfindin-ssin termsof classicalNewtonianmotions.but suchatrempts
eventuallyfailed.Theyfoundthatsuchsrnalllightparticlesbehavedin a waythat
simplyis notconsistentrviththeNeu'tonequations.Let me norvillustratesorne
ofthe experimentaldatathatgaveriseto theseparadoxesandshorvyou how the
scientistsofthoseearlytimesthenusedthesedataro suggestnewequatronsthat
theseparticlesrnightobcy.I wantto stressthattheSchrcidingerequationwasnot
derivedbut postulatedby thesescientists.In fact,to date,rloonehasbeenable
to deriretheSchrcidingerequation.
Fron.rthe pioneeringwork of Braggon ditrractionof x-raysfiom planesof
atomsor ionsin crvstals,it wasknownthatpeaksin the intensityof ditliacted
x-rayshavin,uwavelengthi rvouldoccuratscatteringanglesg determinedbythe
larnousBraggequation:
nt : 24 tinp ( 1 . 2 )
whered is the spacingbetweenneighborin,uplanesof atomsor ions.These
quantitiesareillustratedin Fig. I .I . Therearemanysuchdiffractionpeaks,each
labeledbyadifferentvalueoftheinregern(n - 1,2.3,...).TheBra-s-qformula
canbederivedby consideringwhentwophotons,onescatteringfrom thesecond
planein the figureandthe secondscatteringfrom the third plane,will undergo
constructiveinterference.This conditionis met when the ,.extrapath leneth,'

Scatteringof
two beamsat angle6
from two planesin a
crystalspacedby d.
Thebasicsof ouantummechanics
coveredby the secondphoton(i.e.,the lengthfront pointsA to B to C; is an
integermultipleof thewavelengthof thephotons.
Theimportanceof thesex-rayscatterin_eexperimentsto thestudyof electrons
andnuclei appearsin tl.reexperirnentsof Davissonand Gernter.in 1927,u'ho
scatteredelectronsof(reasonably)fixedkineticenerg)'E frommetalliccrvstals.
Theseworkersfoundthatplotsofthenumberofscatteredelectronsasafunctionof
scatteringangled displayed"peaks"
atangles6 thatobeyedaBragg-likeequation.
The startlingthingaboutthis observationis thatelectronsareparticles.vetthe
Braggequationis basedon the propertiesof waves.An importantobservation
derivedfron.rthe Davisson-Germerexperimentswasthatthe scatteringangles6
observedfor electronsof kineticenerg.vE couldbefit to theBraggni, : 2d sin0
equationif a wavelengthwereascribedto theseelectronsthatu'asdefinedby
) " : h l ( 2 m " E ) t ' 2 . ( 1 . 3 )
whererl. is the massof the eiectronandi is the constantintroducedby Max
PlanckandAlbert Einsteinin theeariy 1900sto relatea photon'senergv,Eto
its frequencyy I'ia .E: /rt,.Theseamazingfindingswereamongthe earliestto
suggestthat electrons,u,hichhad alwaysbeenviewedasparticles,might have
somepropertiesusuallyascribedto waves.That is. asde Broglie suggestedin
1925,anelectronseemsto havea wavelengthinverselyrelatedto itsmomenfum,
andto displaywave-typediffraction.I shouldmentionthatanalogousdiffraction
u'asalsoobservedwhenothersmalllightparticles(e.g.,protons.neutrons,nuclei,
and small atomic ions) were scatteredfrom crystal planes.In all such cases,
Bragg-likediffractionis observedandtheBraggequationis foundto governthe
scatteringanglesifone assignsawavelengthtothescatteringparticleaccordingto
) , : h l Q m E ) 1 t 2 , ( 1 . 4 )
where rl is the mass of the scatteredparticle and /r is Planck'sconsranr
(6.62x l0-27ergs).

Why quantummechanicsis necessaryfor describingmolecularproperties
l,.inm
a c o c- c h a t c
I VisibleI
*.'{fllllrilt-Irffi
frlllill ""'-*,,-onllrll
Anarvsis
lrrn Paschen
Llf Brackett
Theobservationthatelectronsandothersmalllightparticlesdisplayivave-like
behaviorr.vasimportantbecausetheseparticlesarewhatall atomsandmolecules
are madeof. So, if we want to fully understandthe motions and behaviorof
molecules.rvemustbesurethat*e canadequatelydescribesuchpropertiesfor
theirconstitr"rents.BecausetheclassicalNewtonecluationsdonotcontaintactors
thatsr-rggcstwavepropertiesfor electronsor nuclei mo'",ingfreelyin space.the
abovebehaviorspresentedsignificantchallenges.
Anotherproblemthatarosein earlystudiesof atomsandmoleculesresulted
fiom the stLrdyof thephotonsemittedfrom atomsandionsthathadbeenheated
or otherr.iseexcited(e.g.,by electricdischarge).It wasfoundthateachkind
of atom(i.e.,H or C or O) ernittedphotonsrvhosefrequenciesu wereof very
characteristicvalues.An exampleof suchemissionspectrais shownin Fig. I .2
fbr hydrogenatoms.In thetop panel,we seeall of the linesemittedwith thcir
wavelengthsindicatedin nanometers.The otherpanelsshowhorvtheselines
havebeenanalyzed(by scientistsrvhosenarnesareassociated)intopatternsthat
relateto the specilicenergylevelsbetweenwhichtransitionsoccurto emit the
correspondingphotons.
In theearlyattemptsto rationalizesuchspectrain termsof electronicmotlous.
onedescribedan electronasrnovinsabouttheatomicnucleiin circularorbits
suchas shor.vnin Fig. 1.3.A circularorbit wasthoughtto be stablewhenthe
outwardcentrifugalfbrcecharacterizedby radiusr.andspeedu (rr.u2/r) on the
electronperf-ectlircounterbalancedtheinwardattractit,eCoulombforce(Ze2l121
exertedby thenucleusof chargeZ:
Emission
spectrumof atomic
hydrogenwith some
linesrepeatedbelowto
illustratethe seriesto
whichtheybelong.
n.r- ,'t' = Ze-lt
-
.
This equation,in turn, allowsoneto relatethe
CoulornbicenergyZe2lr, andthusto expressthe
(1 . 5 )
kineticenergylrr.ul to the
totalenergyE of anorbitin

Characterizationof
small and large stable
orbits of radii 11 and 12
for an electron moving
a r o u n o a n u c l e u s .
The basicsof quantummechanics
terrnsof theradiusof theorbit:
I - l
E : - r t t . . r -- Z t ' - , r ' : - Z L ' -/ t ' .
f ' ' 1
( 1 . 6 )
The energycharacterizinganorbit ofradiusr. relatil'eto the 6 : 0 reference
of energyatr --+ 3p.becomesmoreandmorenegative(i.e.,lorverandlou,er)asr
becomessmaller.This relationshipbetweenoutu'ardandinu,ardforcesallou's
oneto concludethatthe electronshouldrnovefasteras it movescloserto the
nucleussincer'2: Ze7l(rntr). Howeter.noq'herein thismclclelis a conceptthat
relatesto theexperimentalfactthateachatoll erltitsonll'certainkindsof pho-
tons.lt u'asbelievedthatphotoner.nissionoccurreduficn anelectronr.rrovirrrirr
a largercircularorbitlostenergyandnrovedto a sn.rallercircularorbit.Hor.r'ever.
theNewtoniandynamicsthatproducedtheaboveequationr.vouldallou'orbrtsof
anyradius.andhenceanyenergy.to befollorved.Thus"it wouldappcartlratthe
electronshouldbeableto emitphotonsof anyencrgyasit movedfi'omorbitto
orbit.
The breakthroughthat allowedscientistssuchas Niels Bohr to applythe
circular-orbitmodelto the observedspectraldatainvol','edfirst introducinsthe
ideathattheelectronhasa wavelengthandthatthisu'avelensthi is relatedto
its nromentumby the de BroglieequationL- hlp. The key stepin the Bohr
modelu'asto alsospecifythatthe radiusof the circularorbit be suchthatthe
cilcurnferenceof thectrcle2nrequalaninteger(n)multipleof theu'avelen-gthi.
Only in thiswaywill theelectron'svu'aveexperienceconstructiveinterferenceas
the electronorbitsthe nucleus.Thus,the Bohr relationshipthatis analogousto
the Braggequationthatdeterminesat whatanglesconstructiveinterferencecan
occuris
2trr: n).. (1.7)
BoththisequationandtheanalogousBraggequationareillustrationsof whatwe
call boundaryconditions;theyareextraconditionsplacedon thewavelengthto
producesomedesiredcharacterin theresultantwave(in thesecases,constructile
interference).Of course,thereremainsthe questionof why one must impose
theseextraconditionswhenthe Neu,toniandynamicsdo not requirethem.The
resolutionof thisparadoxis oneof thethingsthatquantummechanicsdoes.
Returningtotheaboveanalysisandusingl,: hlp: hl(mv).2rr :4)',sg
weli astheforce-balanceequationm"r21, : Z e2I 12, onecanthensolvefor the
radii thatstableBohr orbitsobey:
y :1nhl2n)2 /(m"Ze2)
and, in turn. for the velocities ofelectrons in these orbits,
( l 8 )
v : ze2lfuhl2tr). (l.e)

Thesetrvoresultsthenallorvoneto expressthe sumofthe
Coulombpotential(-Ze2 lrl energiesas
g : -!r,,zt r' 1tnlt12t1'.) '
Justas in the Bragg diflraction result,rvhich specifiedat what anglesspecial
high intensitiesoccurredin the scattering,thereare many stableBohr orbits.
eachlabeledby a value of the integerir. Those with small n havesmall
radii. high velocitiesand more negativetotal energies(n.b.,the reference
zero of energycorrespondsto the electronat r : oc, and with v :0). So.
it is the resultthat only certainorbitsare "allowed"that causesonly certain
energiesto occurandthusonly certainenergiesto be observedin the emitted
photolls.
It turnedout thatthe Bohr formulafor the energylevels(labeledby r) of
an electronmovingabouta nucleuscouldbe usedto explainthe discreteline
emissionspectraof rll one-electronatomsandions{i.e..H. He . Li-:. etc.)to
veryhighprecision.in suchaninterpretationof theexperimentaldata.oneclaims
thata photonof energy
h v = R ( t l n i - r l n i ) ( l . l r )
is emittedrvhenthe atomor ion undergoesa transitionfiom an orbit having
quantumnumberni to a lower-energyorbithavingnf. HerethesymbolR isused
to denotethefbllo*ing collectionof factors:
I
R - - n t , Z - e * l l l t r ) t l ' ( 1 . 1 2 )
TheBohrformulafbr energylevelsdid notagreeaswell withtheobservedpattern
of emissionspectrafor speciescontainingmorethana singleelectron.However,
it doesgivea reasonablefit, for example.to theNa atomspectraif oneexamrnes
only transitionsinvolvingthe singlevalenceelectron.The primaryreasonfor
thebreakdorvnof theBohrformulaistheneglectof electron-electronCoulomb
repulsionsin itsderivation.Nevertheless.thesuccessof thismodelmadeit clear
thatdiscreteemissionspectracouldonlybeexplainedby introducingtheconcept
thatnotall orbitswere"allowed".
Onlyspecialorbitsthatobeyeda constructive-
intert-erenceconditionwerereallyaccessibleto theelectron'smotions.Thisidea
that not all energiesrverealloweclbut only certain"quantized"energiescould
occurwasessentialto achievingevena qualitativesenseof agreementr.viththe
experimentalfactthatemissionspectrawerediscrete.
In summary,two experimentalobservationson thebehaviorof electronsthat
werecrucialto the abandonmentof Newtoniandynamicswerethe observations
of electrondiffractionandof discreteemissionspectra.Both of thesefindings
seemto suggestthat electronshavesomewave characteristicsand that these
waveshaveonly certainallowed(i.e.,quantized)wavelengths.
kinetic(lrr.r':) ancl
( 1 . r 0 )

Fundamentalandfirst
overtone notesof a
violinstring.
So.nou,rvehavesomeideawhy the Newtonequationsfail to accountfor
thedynan.ricalnrotionsof lightandsmall;lartrclessuchaselectronsandnuclei.
We seethatextraconditions(e.g..the Braggconditionor constraintson thede
Brogiieu'avelength)couldbeimposedto achievcsomedegreeof agreenrenlu'ith
experimentalobservation.Hou'e'u'er,u'estillarelcftwonderingr.ihattheequations
are that can be appliedto properlydescribesuchmotior.rsand why the extra
conditionsareneeded.It turnsoutthataner.r'kindof equationbasedoncornbining
u'aveandparticlepropertiesneededto bedevelopedto addresssuchissues.These
aretheso-calledSchrddingerequationsto ri'hichwe no'tunrour attention.
As I saidearlier.no onehasvet shownthatthe Schrodi-qerequationfollou,s
deductivelirfrom somen.rorefundamentaltheory.Thatis. scientistsdid notde-
rivethisequatiotr;theypostulatedit. Sorreideaof hou'thescientistso1'thatera
"dreamedup" the Schriidingerequationcanbe hadby exan-rir.ringthc tirre and
spatialdepcndencethatcharacterizesso-calledtravelingu,aves.lt shouldbenoted
thatthepeoplervhoworkedon theseproblcmskneu'a greatdcalaboutu'aves
(e.g.,soundw'avesand.ater'aves)andthc equationstheyobeyed.Moreover.
theykneu'thatu,avescouldsometimesdisplaythe characteristicof quantized
u'avelenqthsor fi'equencies(e.9..fundamentalsandovertonesin soundu'a'"'es).
They'kneu',for example.thatr.r,a'u'esin onedinrensionthatareconstrainedattrvo
points(e.g..a violin stringheldfixedat tu'o ends)undergooscillatorymotjon
in spaceandtimewith characteristicfi'equenciesandu'avelengths.Forexan.rple.
the motionof the violin stringjust nrentionedcanbe describedas havingan
arnplitudeA(r. t ) at a position-r alon-qits len-sthat tinleI gircn by
l(,1-./): ,4(.r-.0)cos(2.'ryr). (1.13)
wherer,is itsoscillationfrequency.Thearnplitude'sspatialdependencealsohas
a sinusoidaldependencegivenby
I (,r.0) : A sin(2r:;l).1. ( 1 . 1 . 1 )
where.),is the crest-to-crestlengthof thervave.Two examplesof suchwaves1n
onedimensionareshownin Fig. 1.4.In thesecases,thestringisfixedatx : 0 and
atx : l, sothe$,avelengthsbelongingto thetwo wavesshownare), : 2L and
),= L.lf the violin stringwerenot clarnpedat ir: I, the wavescouidhave
any valueof ),. Horvever.becausethe string is attachedat ir : l. the allowed
u'avelengthsarequantizedto obey
) " = L l n . ( 1 . 1 5 )
wheren:1.2,3.4,...The equationthatsuchwavesobey,calledthewave
equation.reads
d 2 , 4 ( x , t ) . d 2 A
Lll- (l-
sin(hx/LJ
(r . l 6 )

wherec is thespeedat which thewavetravels.This speeddependson thecom-
positionof the materialfrom which theviolin stringis made.Using the earlier
expressionsfor the-r-andt-dependencesof the wave,A(x,t). we find thatthe
rvavesfrequencyandwavelengtharerelatedby theso-calleddispersionequation:
v: : (c ln):. (r.r7)
or
c : t v . ( 1 . 1 8 )
This relationshipimplies,for example.thatan instrumentstringmadeof a very
stiffmaterial(largec) will produceahigherfrequencytonefor agivenwavelength
(i.e.,a givenvalueof n) thanwill a stringmadeof a softermaterial(smallerc).
For waves moving on the surface of, for example. a rectaneulartwo-
dimensionalsurfaceof lengthsI,, and1,,, onefinds
l(.r. r. /) = sinln,:rr I L.,)sin(n,:ryI L, )cos(2zur) ( 1 . 1 9 )
Hence.the wavesare quantizedin two dimensionsbecausetheir wavelengths
mustbeconstrainedto cause,4(x.y. r) to vanishat_r: 0 and-r: Z. aswellas
at-r,
- 0 and,y: I,, for all timest. Let us now returnto the issueof wavesthat
describeelectronsmoving.
The pioneersof quantummechanicsexaminedfunctionalforms similarto
thoseshownabove.Forexample,fbrmssuchasI : exp[*22 i(ur - .y/i)] were
consideredbecausetheycorrespondto periodicwavesthatevolvein-vanclt under
no external,t'-or l-dependentforces.Noticingthat
(1.20)
andusingthedeBrogliehypothesisX: hlp in theaboveequation.onefinds
( r . 2 1 )
If I is supposedto relateto themotionof a particleof momentump underno
externaltbrces(sincethe waveformcorrespondsto this case),p2 canbe related
to the energyE of the particleby E : p212m. So, the equationfor I canbe
rewrlttenas
(1.22)
J t . J / 2 2 r
; - : - l . l l(/.- , ,/
, l : . 1 . 7 1 . 2 l
- : - p - l , | , a
.1.'- tt /
'ti1=-2,,r('* )' ,,.(1.'- n /
or,aiternatively.
-(*)'#: uo r t 1 l

1 0 Thebasicsof quantummechanics
Returningto thetime-dependenceof l(.r'. /) andusingv : Elh. onecanalso
showthat
( 1 . 2 4 )
which,usingthefirstresult.suggeststhat
( 1 . 2 s )
Thisis aprimitiveformof theSchrcidingerequationthatwervilladdressin much
moredetailbelow.Briefly,what is importantto keepin mind is that the useof
thedeBroglieandPlanclCEinsteinconnectionse,: h/p an<iE : lr,), bothof
whichinvolvetheconstanti. producessuggestiveconnectionsbetrveen
(r.26)
andbet."veen
( 1 . 2 7|
or,alternatively.between
( 1 . 1 8 )
Theseconnectionsbetweenphl,sicalproperties(energyf andmornentump)and
differentialoperatorsaresomeofthe unusualfeaturesofquantum mechanics.
The abovediscussionaboutwavesandquantizedu,avelengthsasrvellasthe
obser"'ationsaboutthe waveequationand differentialoperatorsare not meant
to provideor evensuggesra derivationof the Schrodin_eerequation.Again the
scientistswho inventedquantummechanicsdid not deriveitsrvorkingequations.
Insteadtheequationsandrulesofquantummechanicshavebeenpostulatedand
designedto be consistentu'ith laboratoryobservations.My studentsoften find
thisto bedisconcertingbecausetheyarehopingandsearchingfor anunderlying
fundamentalbasisfrom vvhichthebasiclarvsof quantummechanicsfollou,log-
icall1,.I try to remindthem thatthis is not hou'theoryu,orks.lnstead.oneuses
experimentalobservationto postulatea rule or equationor theory,.andonethen
teststhetheoryby makingpredictionsthatcanbetestedby furtherexperiments.
Ifthe theoryfails, it mustbe "refined",
andthisprocesscontinuesuntil onehas
a betterandbettertheory.In this sense.quantummechanics,with all of its un-
usualmathematicalconstructsandrules.shouldbe viewedasarisingfrom the
imaginationsof scientistswho tried to inventa theorythatwasconsistentwith
experimentaldataandwhich couldbe usedto predictthingsthatcouldthenbe
testedin the laboratory.Thusfar, thistheoryhasprovenreliable,but,of course,
t h 1 d . 4 r l t : J : A
' = /
, t ,
: -
; / , . .
/ h d A
i ( ^ I ' : E i t '
t l r / d l
,(+)# andE
r , 1 / t 1 : d :
D- anc
2:r) dt:
7.' and- , (:-' + :1T / .1.

TheSchrodingerequationand itscomponents
we areahvayssearchingfor a "new and improved"theory that describeshow
smalllightparticlesmove.
If it helpsyou to be moreacceptingof quantumtheory,I shouldpoint out
thatthe quantLrmdescriptionof particleswill reduceto the classicalNewton
descriptionundercertaincircumstances.In particular,whentreatingheavypar-
ticles(e.g.,macroscopicmassesandevenheavieratoms),it is oftenpossibleto
useNer,vtondynamics.Briefly.rvewill discussin moredetailhow the quantum
andclassicaldynamicssometimescoincide(in which caseone is freeto use
the simplerNeu,tondvnamics).So. let us now moveon to look at this strar.rge
Schrodingerequationthatlvehavebeendi-eressingaboutfor solong.
1.2 TheSchrodingerequationand its components
It hasbeenwell establishedthatelectronsmovingin atomsandmoleculesdo
notobeytheclassicalNewtonequationsof motion.Peoplelongagotriedto treat
electronicnlotionclassicallv.andtbundthatfeaturesobservedclearlyin erperi-
mentalmcasurementssimplywerenotconsistentwithsuchatreatment.Attentpts
r,veremadeto supplcrnenttheclassicaleqr-rartionsrvithconditionsthatcouldbe
usedto rationalizesuchobservations.For eraniple.earlvrvorkersrequiredthat
theangularmomentllmL : r x p beallowedto assumeclnl,vintegermultiplesof
l/22 (lvhichis ofienabbrcviatedasfi),whichcanbeshorvnto becquivalenrto
theBohrpostulatetti : 2:rr. Hor,vever.untilscientistsrealizeclthata ner',,setof
lar.ls.thoseof cluantumrlechanics.appliedto lightrnicroscopicparticles.a rvide
gullexistedbetr.leenlaboratoryobserlationsof molecule-lcvelphenomenaand
thcequationsuscdto describesuchbehavior.
Quanturnrnechanicsiscastin a languasethatis notfarniliarto moststudents
of chemistryrvhoareexarrininsthesubjectfbr thefirsttimc-.Its nrathernatical
contentandhorvit relatcsto experimentalmeasurementsbothrecluirea-grc-atdeal
of eftortto mastcr.With thesethoughtsin mind.I haveorganizedthisntaterialin
a mannerthatfirstprovidesa briefintroductionto thetwo prirnaryconstructsof
qLlantummechanicsoperatorsandwavefunctionsthatobeyaSchrodingerequa-
tion.Next.I dcmonstratctheapplicationof theseconstructsto severalchemically
relevantmodelproblents.By learningthesolutionsof theSchrodingereqLlatlon
fbr a f'ewmodelsystems,the str"rdentcan betterappreciatethe treatmentof the
fundamentalpostulatesof quantummechanicsaswellastheirrelationto experi-
mentalrreasurementfor whichthewavefunctionsof theknolvnmodeloroblems
off!r inrportuntinrerpr,.'rations.
1.2.1Ooerators
Eachphysicallymeasurablequantityhasa correspondingoperator,The
eigenvaluesoftheoperatortelltheonlyvaluesofthecorrespondingphys-
icalpropertythat can be observed.
t l

Any experimentallymeasurablephysicalquantityF (e.g.,energy.dipolemo_
ment.orbitalangularmomentum.spinangularmomentum,linearmomentum,
kineticenergy)hasaclassicalmechanicarexpressionin termsof thecartesianpo_
sitions{q;} andmomenta{p;} of theparticlesthatcomprisethesystemof interest.
Eachsuchclassicalexpressionis assigneda correspondinsquantummechanical
operatorF formedby replacingthelp1) in the classicalform by the differentiar
operaror-i/1 0/0q 1ar.rdreavingthecoordinates97thatappearin F untouched.
Forexample.theclassicarkineticenergyof y'y'particles(u,ithmassesr?1)movlnir
in a potentialfield containingboth quadraticandrinearcoordinate-def"nd.n.e
canbewrittenas
r:,I,
lt'i/2u,,+t/2k(q1-,tl),+ t (q,- q),)l
Thequanturnmechanicaloperatorassociatedwith this .F is
Suchanoperatorwouldoccurwhen,for example,onedescribesthesumof'the
kineticenergiesof a collectionof particles(thef,:, ,,(7ti/2,,11rerm).plusrhe
sumof "Hooke's
Lavu'''parabolicpotentialstthe lfZy,,=t.t k(qr_ q!t'721,ana
(the lastterm in F) the interactionsof the particles*itn ln .*r.rnaily appried
fieldwhosepotentiarenergyvarieslinearryastheparticlesmoveaivayfrom their
equilibriumpositions{4,0}.
Letustry moreexampres.Thesumof the:-componentsof angularmomenta
(recallthatvectorangurarmomenfurnL is definedasL : r x p; ota collection
of ly'particleshasthefollowing classicalexpression:
t:
f GiPvt-)'tP,i),
-/= I ..''
andthecorrespondingoperatoris
t :,I,
L*,# * )r(,'- q;')'+t'(q,-'ti')]
t : -t,.,:.Iu(',*
- r',+)
p - -' -
./--
( 1 . 2 9 )
(I . 3 0 )
(r.31.)
(r.32)
(1.33)
If onetransformsthesecartesiancoordinatesandderi'ativesinto polar coordi_
nates,theaboveexpressionreducesto
F - - i l t f
a
i?' 3Qi
The r--componentof the dipole momentfor a coilectionof 1y'particleshas a
classicalform of
Z1exi, (r.34)

for whichthequantttmoperatoris
1 2
*'hereZ1eis thechargeon theTthparticle.Noticethatin this case,classicaland
quantumformsareidenticalbecause-Fcontainsno momentumoperators.
Themappingfrom F to F is straightforwardonly in termsof Cartesiancoor-
dinates.To mapa classicalfunction.E,givenin termsof curvilinearcoordinates
(evenifthev areorthogonal),into its quantumoperatoris not at all straightfor-
ward.The mappingcanalwaysbe donein terrnsof Cartesiancoordinatesafter
whicha transtbrmationof theresultingcoordinatesanddifferentialoperatorsto
a curvilinearsystemcanbeperformed.
Therelationshipof thesequantummechanicaloperatorsto experimentalmea-
surementlies in the eigenvaluesof the quantumoperators.Eachsuchoperator
hasa correspondingeigenvalueequation
F x ,: u , x , (r.36)
in whichtheX7arecalledeigenfunctionsandthe(scalarnumbers)cv;arecalled
eigenvalues.All sucheigenvalueequationsareposedin termsof agivenoperator
(F in thiscase)andthosefunctions{X/} thatF actson to producethefunction
backagainbLrtmultipliedbv a constant(theeigenvalue).Becausetheoperator
F usuallircontainsdifferentialoperators(comingfrom the momentum).these
equationsaredifferentialequations.TheirsolutionsX, dependonthecoordinates
thatF containsasdiflerentialoperators.An examplewill helpclarif,vthesepoints.
Thediffercntialoperatordf d.r'actson rvhatfunctions(of_r,)to generatethesame
functionbackagainbut multipliedby a constant?Theansweris functionsof the
form exp(rl-r,)since
F : I Z , e x i , (r.3s)
( 1 . 3 7 )
So.we saythatexp(a_r,)is an ei-eenfunctionof dldy anda is thecorresponding
eigenvalue.
As I will discussin moredetailshortly,theeigenvaluesof theoperatorF tellus
theonlt,valuesofthe physicalpropertycorrespondingto theoperatorF thatcan
be observedin a laboratorymeasurement.SomeF operatorsthatwe encounter
possesseigenvaluesthatarediscreteor quantized.Forsuchproperties.laboratory
measurementwill resultin only thosediscretevalues.Other F operatorshave
eigenvaluesthatcantakeon a continuousrangeof values;for theseproperties,
laboratorymeasurementcangiveanyvaluein this continuousrange.

1.2.2Wavefunctions
The eigenfunctionsof a quantum mechanicaloperatordependon the
coordinatesupon which the operatoracts.The particularoperatorthat
correspondsto the total energyof the systemis calledthe Hamiltonian
operator.The eigenfunctionsof this particularoperatorare calledwave
functions.
A specialcaseofan operatorcorrespondingto a phvsicalll,nreasurablequan-
titv is the HamiltonianoperatorH thatrelatesto thetotalenercyof the system.
The energyeigenstatesofthe svstemW arefunctionsofthe coordinates1g;1
thatH dependson andof time l. The functioniV(r7;.1)ll: W-V givesthe
probabilitydensitvfor observingthecoordinatesat thevaluesq; at tirnel. For
a manv-particlesystemsuchasthe H2o molecule"the u'avefunctiondepends
on manycoordinates.ForH2O.it dependson the.x..r..and: (or r. 0. andS1
coordinatesof thetenelectronsandthe.r-..r,.and: (orr. 0. and(t) coordinatesoi'
theoxygennucleusandofthe twoprotons;a totalof39 coordinatesappearin v.
ln classicalmechanics.thecoordinatcsq7 andtheircorrespondinsmomenta
p.i arefunctionsof tinre.Thestateof thesystemis thendescribedbl,spccifvine
qj(l) andp 1(tl.In quantumntechanics.theconceptthat17;isknor,"'nasa funr:tion
of tirneis replacedby theconceptof theprobabilitvdcnsitvfor 1indingQi ar ?
particular"'alueata particulartime lv(r7r.t)lr. Knorvledgeof thccorresponding
momentaasfunctionsof timeis alsorelinquishedin quantumnrechanics:lgain.
onlyknou'ledgeof theprobabilitvdensiryforfinding7r,r.vithanyparticularr,,aluc
at a particulartime/ remarns.
The Hamiltonianeigcnstatesareespeciallyirnportantin chenristrybecause
manyof thetoolsthatchemistsuseto studymoleculesprobetheenergystatesof
themolecule.Forexample,nrostspectroscopicmethodsaredcsignedtodeterrnine
u,hichenergystateamoleculeis in. However.thereareotherexperimentalmelh-
odsthatmeasureotherproperties(e.g.,the:-cornponentof angularmonrentun.r
or thetotalangularmomentum).
As statedearlier,if thestateof somemolecularsvstemis characterizedb1'a
wavefunctionw thathappensto be an eigenfunctionof a quanlummechanical
operatorF, one can immediatelysay somethingaboutwhat the outcome$,ill
be if the physicalpropertyF correspondingto the operatorF is measured.In
particular,sir.rce
Fxt : )",x1. (r.38)
wherei 7isoneof theeigenvaluesof F. weknowthatthevaluei7 wili beobserved
if thepropertyF ismeasuredwhilethernoleculeisdescribedbythervavefuncrion
'4t: X.i.ln fact,oncea measurementof a physicalquantityF hasbeencarried
outanda particulareigenvalue).i hasbeenobserved-thesystem'srvavefuncrion
v becomestheeigenfunction1, thatcorrespondsto thateigenvalue.Thatis,the

actof makingthemeasurementcausesthesystem'swavefunctionto becomethe
eigenfunctionof thepropertythattvasmeasured
Whathappensif someotherpropertyG, r.vhosequantummechanicaloperator
is G is measuredin sucha case?Weknou'from whatwassaidearliertharsome
ei_genvaltrepr of theoperatorGlvill beobservedin themeasurement.But.will
themolecr-rleiwavefunctionrenrain,afterG is measuredtheeigenfunctionof
F, or will themeasurementof G causeV to bealteredin a waythatmakesthe
molecule'sstateno longeran eigenfunctionof F? It turnsout that if the two
operatorsF andG obeythecondition
F G = G F (1.39)
then.whenthepropertyG is measuredthewavefunction'Q: Xtrvili remain
unchanged.Thispropertv.thattheorderofapplicationofthe two operatorsdoes
notmatter.is calledcontmutation:thatis,we savthetu,ooperatorscommuteif
theyobeythispropertv.Letusseehowthispropertyleadsto theconclusionabout
V remainingunchangedif thetr,vooperatorscomnlute.In particulaqwe apply
theG operatorto theaboveeigenvalueequation:
C F7. ,: G 1iXi. ( r..10)
andNext.rveusethecomntutationto re-writctheleti-handsideof thisequation.
usethefactthati, isa scaiarnurnbertothusobtaln
F G 7 i : r . Q 7 , ( 1 . - 1 l t
So. nor.vwe scc that (G;1,) itselfis an ei-uenfunctionof F havingeigenvalue
ir. So, unlessthereare l.norethanoneeigenfirnctionsof F correspondingto
theeigenr.aluci; (i.c..unlessthiscigenvalueis degenerate).GX, n.rustitselfbe
proportionalto 21r.Wewritethisproportionalirvconclusionas
G X , : l t i x , . ( 1 . : 1 2 )
which meansthat X, is alsoan eigenfunctionof G. This. in turn. meansthat
measuringthe propertyG rvhilethesystemis describedby the wavefunction
tp : Xidoesnotchangethewavetunction:it remainsX,.
So.lvhenthe operatorscorrespondingto two physicalpropertiescommLlre,
onceonemeasuresoneof the properties(andtliuscausesthe systemto be an
eigenfunctionofthat operator).subsequentmeasurcmentofthe secondoperator
will (iithe eigenvalueof thefirstoperatoris not degenerate)producea uniqtLe
eigenvalueofthe secondoperatorandwill notchangethesystemwavefunction.
If the two operatorsdo not commute.one simplycan not reachthe above
conclusions.In suchcases.measurementof the propertycorrespondingto the
firstoperatorrvill leadto oneofthe eigenvaluesofthat operatorandcausethe
systemlvave function to becomethe correspondingeigenfunction.However,
1 5

t o Thebasicsof quantummechanics
subsequentmeasurementof the secondoperatoru'ill producean eigenvalueof
thatoperator.but thesystemu'avefunctionwill bechangedto beconreancrgen-
functionofthe secondoperatorandthusno longertheeigenfunctionofthe first.
1.2.3TheSchrodingerequation
Thisequationis an eigenvalueequationfor the energyor Hamiltonian
operator;its eigenvaluesprovidethe only allowed energy levelsof the
sysrem.
The time-depe ndent equation
lfthe Hamiltonianoperatorcontainsthetimevariableexplicitly,onemust
solvethe time-dependentSchrodingerequation.
Beforemovingdeeperintounderstandingwhatquantummechanics"means".
it is usefulto learnhorvthe wave functionsV are found by applyingthe ba-
sicequationof quantun.rmechanics.the Schrodingerequation.to a few exactly
solublemodelproblems.Knorvingthe solutionsto these"easy"
vet chernically
veryrelevantmodelsr.l,illthenfacilitatelearninsmoreof thedetailsaboutthe
structureof quantummechanics.
TheSchrodingerequationis a differentialequationdependingontirneandon
allofthespatialcoordinatesnecessarl,todescriberhesystemathand(thirt1,-nine
for theH2Oexamplecitedabove).lt is usuallyn,ritten
HtU:i716ry1'6,. (1.43)
wherev(qi, t.)is theunknownwavefunctionandH is theoperatorcorrespond-
ing to the total energyof the system.This operatoris calledtl.reHar.niltonian
and is formed,as statedabove,by first writing down the classicalmechanical
expressionfor the total energy(kineticplus potential)in Cartesiancoordinates
and momentaand then replacingall classicalmomentap; by their quantum
mechanicaloperatorspj : -ih 0lSq.i.
FortheH2o exampleusedabove,theclassicalmechanicalenergyof allthirteen
particlesis
F _ l / r - e - - 2 , " ' t't - /_J t.t.t ? ,,"
I
s . I p ;
+lr^-
r - f p j- +lr*
-r-rir - Z'Zr'i I' - ' - ?
r u hI '
(r.44)
wherethe indicesi andj areusedto labelthe l0 electronswhose30 Cartesian
coordinatesarelqil anda andb labelthethreenucleiu,hosechargesaredenoted
{Zo}, andwhosenine cartesiancoordinatesare lqo}.The electronandnuclear

massesaredenotedrr. and {n,,}, respectively.The correspondingHamiltonian
operatoris
NoticethatH isasecondorderdifferentialoperatorin thespaceofthe39Cartesian
coordinatesthatdescribethepositionsofrhe renelectronsandthreenuclei.lt is
a secondorderoperatorbecausethemomentaappearin thekineticenergyaspr2
andp;,and thequantummechanicaloperatorfor eachmomentump : _ f ttlI 0q
is of first order.TheSchrodingerequationfor the H2o exampleat hand then
reads:
w
;l r!
y : ln-
A t
TheHamiltonianin thiscasecontains/ nowhere.An exampleof a casewhere11
doescontain/ occurswhenanoscillatingelectricfieldE cos(ror)alongthe-r-axrs
interactswith theelectronsandnucleianda term
)
Z,rX,,f.cos((r/) -
f e;r7Ecos(rr-rr) (1.47)
is addedto theHamiltonian.Here,xo and.,videnotethe.r coordinatesof theath
nucleusandthe7th electron.respectivelv.
The time-i ndepe ndent equatio n
lf the Hamiltonianoperatordoes not containthe time variableexplicitly,
onecansolvethe time-independentSchrodingerequation.
In caseswherethe classicalenergy,andhencethe quantumHamiltonian,do
notcontarntermsthatareexplicitlyrimedependent(e.g..interactionswith time
varying externalelectricor magneticfields rvouldadd to the aboveclassical
energyexpressiontimedependentterms),theseparationsof variablestechniques
canbeusedto reducetheSchrodingerequationto a time-independentequation.
In suchcases,H is not explicitlytime dependent,so one can assumethat
v (.qi ' t) is of theform (n.b.,thisstepis anexampleof theuseof theseparations
of variablesmethodto solvea differentialequation)
1 7
H=El*#.:l*-+Tl
.+[-*#.:+'#l .'15)
t l-Ai- - 1- "' - - Z,e:
' 7
1 2 n , d q ; ? ? r , , ? , . , "
- t l _ l a :
* 1 l Z , Z n e :' l l 2 n , ' d q . ; 2 ? r r . h
( 1.16)
V( q, ,t ) = W ( qi) F( t ) ( l.4 E)

Substitutingthis"ansatz"
intothetir.ne-dependentSchrodin_serequationgives
V( q1)it t0 F l'd t: F Q)HV(qtt. (1 .4 9 )
Dividingby V(q 11F(t)thengives
F t t i ht ) F l i t r t : W i [ H , ! ( 4 i) ] . ( 1 . 5 0 )
SinceF(l ) is onlya functionof timet. andtU(q1)is onlya functionof thespatial
coordinateslq1], andbecausetheleft-handandriglit-handsidesmustbe equal
for all valuesof t andof lq il, boththe left- andright-handsidesmustequala
constant.lf thisconstantis calledE. thetwo equationsthatareernbodiedin this
separatedSchrodingerequationreadasfollou,s:
H V ( q , ) : E V ( q , ) .
i f i d F ( t ) l d t : E F ( t ) .
if ld F (r)ld t: EF (t)
F(t) = gapl-i E1171,
( l . 5 l)
il.sl)
Thefirstoftheseequationsiscalledthetin-re-independentSclrrodingerequatron:it
isaso-calledeigenr,alueequationin u'hichoneisaskedtofindfunctionsthat1,ield
a constantmultipleof ther.nselveswhenactedon by the Hamillonianoperator.
SuchfunctionsarecalledeigenfunctionsofH andthecorrespondingcoustalltsare
calledcig.enraluesof H. Forexample.if H uereof'rheform(_ li '2.lt;t:, dQ: -
H" tlrenfunctionsof thc form exp(inrQ)wouldbeeigenfunctionsbecause
I l i d : I l n : h : l
, -
- - l e x p ( i , , i @ ) : l * f c r p t i i r r d , ) . l l . - i . l )
I z t r a 6 , 1'
[ : , r ,J
Inthiscase.rr 2tt2
12tvtistheeigenvalue.Inthisexample.theHarriltoniancontains
the squareof an angularmomentumoperator(recallearlierthatu,'eshou'edtlre
:-cornponentof angularmomentumis to equal-ih dldQ).
Vhen the Schrcidingerequation can be separatedto -seneratea time-
independentequationdescribingthespatialcoordinatedependenceofthe u'ave
function,theei_eenvalueE mustberetumedto theequationdeterminingF(t) to
find thetime-dependentpartof thewavefunction.By solving
onceE is known.oneobtains
andthefull wavefunctioncanbewrittenas
v(qi.r): w(qi)exp(-i Et ltt). (1.56)
Forthe aboveexample,thetime dependenceis erpressedby
( 1 . 5 4 )
( 1 . 5 5 )
F(r): exp(-irlnt2ti1zll1itt1. (l.s7)

TheSchrodingerequationand itscomponenrs
In summary,wheneverthe Hamiltoniandoesnot dependon time explicitly,
onecansolvethetime-independentSchrodingerequationfirstandthenobtain
rhetime dependenceasexp(-i El/fi) oncethe energyE is known.In thecase
of rnolecularstructuretheory,it is a quite dauntingtaskevento approximately
soivethe full Schrodingerequationbecauseit is a partialdifferentialequarion
dependingon all of thecoordinatesof theelectronsandnucleiin themolecule.
For this reason.thereare variousapproximationsthat one usuallyimplements
whenattemptingto studymolecularstructurer.rsingquantummechanics.
The Born-O ppen heime r approxi m ation
one of themostimportantapproximationsrelatingto applyingquantummechan-
icsto moleculesis knownasthe Born-oppenheimer(Bo) approximation.The
basicideabehindthisapproximationinvolr.esrealizingthatin thefull electrons-
plus-nucleiHamiltonianoperatorintroducedabove,
t v
(r.58)
the tirnescalesivith which the electronsand nucleimoveare generallyquite
difl'erent.In particular.the heavynuclei(i.e.,evena H nucleusweighsnearly
2000timeswhatanelectronweighs)move(i.e.,vibrateandrotate)moreslowly
thandothelirrhterelectrons.Thus.r','cexpecttheelectronsto beableto ..adjust,'
theirmotionsto themuchrrroreslowlymo'n,ingnuclei.Thisobservationmotivates
r.rsto solvethe Schrcidingerequationtbr the movementof the electronsin the
presenceof fixed nucleias a vvilvro representthe fully adjustedsrateof the
electronsat anyfixedpositionsofthe nuclei.
TheelectronicHamiltonianthatpertainsto themotionsof theelectronsin the
presenceof so-calledclarnpednuclei.
H : t l - , i r ' -- l - - " '
? 1 z n ti t q : 2 ? ' ' , ,
* , - l _ f i r ; r r _ l
" |
2nr,,;)q,] 2
H : - l - ' i " - * 1 t ' ' '
-
I
l r t t , : t , 1 : ) L , " . ,
- ' t a l
? , , " 1
7 7 . :
..-
z!/ zxc
';
t'u.h
- r'Z{ lt / , , , l '
( 1 . 5 9 )
producesasitseigenvalues.throughtheequation
HO1(q1 i rl,) : E rkl,,),!.t(q,I q,). ( 1 . 6 0 )
energresErGl,) that dependon wherethe nuclei are located(i.e.,the {r7,,}
coordinates).As itseigenfunctions.oneobtainswhatarecalledelectronicwave
functionsltLrkti I r7,,))whichalsodependon wherethenucleiarelocated.The
energiesExQ") arewhatweusuallycallpotentialenergysurfaces.An example
of sucha surlaceis shoivnin Fig. 1.5.Thissurfacedependson two geometrical
coordinates{r7,,}a'd is a plotof oneparticulareigenvalueEL@,) vs.thesetwo
coordinates.

20 Thebasicsof quantummechanics
fransrtionstructureA
Minimum for
product A
- 0.,5
Secondorder n
saddlepoint
0.5
ralley-ridgc
inflectionpotnt
Minimum for reactant
Although this plot has more informationon it than we shall discussnou'.
a few featuresareworthnoting.Thereappearto be tlireeminima(i.e..points
r,l,herethe derir"ativesof -87rvith respectto both coordinatesvanishandu'here
the surfacehaspositivecurvature).Thesepointscorrespond"aswe u'ill seeto-
u,ardthe end of this introductorymaterial,to geometriesof stablet.nolecular
structures.The surfacealsodisplaystwo first ordersaddlepoints(labeledtran-
sitionstructuresA andB) thatconnectthethreemininra.Thesepointshavezero
first derivativeof E.r with respectto both coordinatesbut haveone direction
of negativecurvature.As we will show later,thesepoints describetransition
statesthatplay crucialrolesin thekineticsof transitionsamongthethreestable
geometries.
Keepin mind thatFig. 1.5showsjust oneof the tJ surfaces;eachmolecule
hasa ground-statesurface(i.e.,the onethat is lowestin energ.v)aswell asan
infinitenumberofexcited-statesurfaces.Let'snou'returnto ourdiscussionofthe
BO modelandaskwhatonedoesonceonehassuchanenergysurfacein hand.
The motions of the nuclei are subsequently,within the^Bo model. as-
sumedto obey a Schrodingerequationin u'hich L"{-(h'l2nt,,)d!lq;-l
1l2l,n ZoZ6e2f ro.6]* Er@) definesarotation-vibrationHarniltonianfor the
particularenergystateE5 ofinterest.TherotationalandI'ibrationalenergiesand
wavefunctionsbelongingto eachelectronicstate(i.e.,for eachvalueof the in-
dex K in Ex@)) arethenfoundby solvinga Schrodingerequationrvithsucha
Hamiltonian.
This BO model forms the basisof much of how chemistsview tnolecular
structureandrnolecularspectroscopy.For exarnple,asappliedto forrraldehyde
H2C:O, we speakof the singletgroundelectronicstate(with all electronsspin
paired and occupyingthe lowestenergyorbitals)and its vibrationalstatesas
Two-dimensional
potentialenergysurface
showinglocalminima,
transitionstatesand
pathsconnectingthem.
SecondordersaddlePotnl

Motionof a particlein onedimension
nell asthe n --+z* andir --->lr*electronicstatesandtheirvibrationallevels.
Althoughmuch morewill be saidabouttheseconceptslaterin this text.the
studentshouldbe arvareof the conceptsof electronicenergysurfaces(i.e..the
lExe,)]) andthevibration-rotarionstatesthatbelongro eachsuchsurface.
Having beenintroducedto the conceptsof operators,wave functions.the
Hamiltoniananditsschrodin-eerequation,it is importantto nowconsiderseveral
examplesof theapplicationsof theseconcepts.Theexamplestreatedbelorvwere
chosento providethereaderwith vaiuableexperiencein solvingtheSchrodinger
equation:theywerealsochosenbecausetheyform then]ostelementarychemical
modelsof electronicmotionsin conjugatedmoleculesandin atoms,rotationsof
linearmolecules,andvibrationsof chemicalbonds.
1.3 Yourfirst applicationof quantum mechanics- motion
of a particlein one dimension
Thisis a very importantproblemwhose solutionschemistsuseto model
a wide varietyof phenomena.
Let'sbeginby examiningthe motionof a singleparticleof mass,n ln one
directionwhichrvewill call"r rvhiler-rnderthe influenceof a potentialdenoterl
Z(,r). The classicalexpressionfor the total energyof sucha systemis .E:
p212n -t V(x), rvherep is the momentumof the particlealongther-axis.To
focuson specificexamples,considerhorvthisparticlewouldmoveif I/(.r.1il'ere
of theformsshor.vnin Fig. I .6.wherethetotalenergyE isdenotedbytheposition
ofthe horizontalline.
1.3.1Classicalprobabilitydensity
I wouldlikeyouto imaginewhattheprobabilitydensitywouldbefor thisparticle
movingwith totalenergyE andwith I/(,r) varyingasthe threeplotsin Fig. L6
21
Three
characteristicpotentials
showingleftand right
classicalturningpoints
at energiesdenotedby
thehorizontalIines.

illustrate.Toconceptualizetheprobabilitydensitl'.imaginetheparticleto havea
blinkinglan.rpattachedto it andthinkof thislampblinkingsay100timesftrreach
timeintervalit takesfortheparticleto completea full transitfromitslefi turning
pointto its right turningpoint andbackto theformer.Theturningpoints,r-sand
-rRarethepositionsatwhichtheparticle,if it weremo'"'ingunderNe'ton'siau's.
wouldreversedirection(asthemomentumchangessign)andturnaround.These
positionscanbe foundby askingwherethe momentumgoesto zero:
0 : p : ( 2 m ( E - I ' ( . t ) ) rr . ( 1 . 6 i )
Thesearethepositionsrvhereall ofthe energyappearsaspotentialenergvE :
I/(.x) andcorrespondin theabovefiguresto thepointsrvherethedarkhorlzontal
linestouchthe I'(r)plots assho'uvnin thecentralplot.
Theprobabilitydensityat anyvalueof -r:representsthe fractionof'tirllethe
particlespendsat thisvalueof ..r-(i.e..u'ithin.r and,t * r/.r).Think of forrrling
thisdensity'by allowingtheblinkinglantpattachcdto theparticleto shedlight
on a photo-eraphicplatethat is exposedto tliis light for manyoscillationso1-
theparticlebet'een-r1and;rp.Alternatively.onecanexpressthis probability
amplitudeP(r) by dividingthespatialdistancer/.r"bythevelocitl'oftheparticle
attlienointr:
P(-t): (2m(E- I"(.r))
I rrrrrir ( i . 6 2 )
BecauseE is constantthroughouttheparticle'smotion.P(,t)tl'ill be smallat,t
valuesu,heretheparticleis mo'u'ingquickly(i.e..'ui'herel/ is lou')andri'ill behigh
wheretheparticlemovesslowly1r,r,heretr/is high).So.thephotographicplate
u'ill shou,a brightregionwhereI/ is high(becausetheparticlemovesslou'lyin
suchregions)andlessbrightnesswhere tr'is lou'.
Thebottomlineis thattheprobabilitydensitiesanticipatedby analyzingthe
classicalNewtoniandynamicsof thisoneparticle'uvouldappearasthehistogran.r
plotsshownin Fig.1.7illustrate.Wheretheparticlehashighkineticenergy(and
thuslowerZ(.lr)),it spendslesstimeandP(.t) is small.Wheretheparticlemoves
Classical
ty plotsfor the
threepotentialsshown
in Fig.1.6.

Motionof a particlein one dimension
slorvly.it spendsmoretime and P(.r) is larger.For the plot on the ri_eht,/(,r)
is constantwithin the "box".
so the speedis constant.hencep(.r) is constant
fLrrall .vvaluesrvithinthisone-dimensionalbox.I askthatyoukeepthesepiots
in nrind becausethey are very difrerentfrom what one finds when onesolves
theSchrcidingerequationfor thissameproblem.Also pleasekeepin mindthat
theseplotsrepresentwhat oneexpectsif theparticleweremovingaccordingto
classicalNewtoniandynamics(w,hichweknowit is notl).
1.3.2Ouantumtreatment
To solvetbr the quantLlmmechanicalwavefunctionsandenergiesof this same
problem,wefirstwritetheHamiltonianoperatorasdiscussedaboveby replacing
n Ov -lll(l /Ll.".
h' ,t:
I l : - ' - - l + l ' r r r
lilt tl.-
We thentry to find solutionsry'(.r)to Hrlr : ErL thatobeycertainconditions.
Theseconditionsarerelatedro thefactthatiry'(.r)12is supposedto betheprob-
abilitydensityfbr findinetheparticlebetrveen-r and.r * r/.r.To keepthingsas
simpleaspossible,let'sfocuson the"box" potential/ shorvnin therightside
of Fig.1.7.Thispotential.expressedasa functionof ,r is: V(x: cc for-r < 0
andfbr,r > L: I/1.r1: 0 fbr.r-between0 andl.
Thefactthattr/is infinitefor-r < 0 andfor.r > l. andthatthetotalenergyE
mustbefinite.saltsthat/r mustvanishin thesetworegionsdt : 0 for_r< 0 and
forx > l). Thisconditionmeansthattheparticlecannot accessthesereslons
wherethepotentialis infinite.Thc secondconditionthat.nvemakeuseof is that
/(-t ) mustbecontinuous:thisrrreansthatrheprobabilityof theparticlebeingrt,r
cannotbediscontinuouslyrelatedto theprobabilityofit beingata nearbypornr.
1.3.3Energiesandwavefunctions
Thesecondorderdift-erentialequation
il tlllr
-
^
-;-. -t tr/r)tlr: Etlt
l n t d r '
(1.6.+)
hastwo solutions(becar.rseit is a secondorderequation)in theregionbetrveen
" r : 0 a n d x : L :
/ :sin(l.r) and ry':cos(,t-r). rvhereAisdefinedask :(2mEltt:1|i1. (1.65)
Hence.themostgeneralsolutionis somecombinationof thesetrvo:
23
( 1 . 6 3 )
ty': .1sintA.r)+ Bcosrfr.r'.y. ( 1 . 6 6 )

z 4 Thebasicsof quantummechanics
Thefactthatry'mustvanishat,v:0tn.b..r/rvanishesforx< 0andiscontinuous.
soit mustr.'anishat tlrepoint.r :0)means thattherveithtinganrplitudeof the
cos(Ax)termmustr,'anishbecausecos(Ar): I at-t : 0. Thatis.
B = 0 . ( 1 . 6 7 )
Theamplitudeof thesin(rt,r)rermis notaffecredby theconditionthat{r vanish
ot.r : 0. sincesin(t.r)itselfr.anishesat,r : 0. So.nou,weknou,that{r is really
of theforn:
/(.r):.{ sin (tr-). ( 1 . 6 [ i )
Theconditionthat1/ralsovanishat-v: I hastrvopossibleimplications.Either
.4:0 or fr mustbe suchthatsin(Al):0. TheoptionI :0 wouldleadto
ananswerry'thatvanishesat all valuesof -r andthusa probabilitythatvanishes
everywhere.Thisis unacceptablebecauseit rvouidimplvthattheparticleisnever
observedanywhere.
Theotherpossibilityis thatsin(ft1): 0. Let'sexplorethisansrverbecausert
offersthe first exanrpleof energyquantizationthat you haveprobablyencoun-
tered.As youknow.thesinefunctionvanishesat integralmultiplesof :r. Hence
AL mustbesomemultipleof :r: let'scalltheintegern andu'riteZA : rz (using
thedefinitionof fr) in the form:
L t } n E l l i t t ) : n n .
Solving this equation for the energy f , we obtain:
( r . 6 9 )
E : n1r2rtl(2mt). (r.70)
This resultsaysthat the only energyvaluesthat are capableof grving a wave
functionry'(x)thatwill obeytheaboveconditionsarethesespecificE r.alues.In
otherwords,notall energyvaiuesare"allowed"
in thesensethattheycanproduce
ry'functionsthatarecontinuousandvanishin regionswhere z(x) is infinite.If
oneusesanenergyE thatis not oneofthe allowedvaluesandsubstitutesthis E
intosin(*;r),theresultantfunctionwill notr,anishatx : L.I hopethesolutionto
thisproblemremindsyouof theviolin stringthatwediscussedearlier.Recallthat
theviolin stringbeingtieddownatx : 0 andat-r : z gaveriseto quantization
of thewavelengthjust astheconditionsthatry'becontinuousat;r-: 0 ar.rdx : Z
gaveenergyquantization.
Substitutingk : nn /L into ry': I sin(k,r)gives
VQ)= Asin(nrx/L) ( r . 7 r )
Thevalueof I canbe foundby rememberingthat lq/l2is supposedto represent
theprobabilitydensityfor findingtheparticleatx. Suchprobabilitydensitiesare

supposedto be normalized.rneaningthattheirintegraloverall x valuesshould
amountto unity.So,we canfind I by requiringthat
where the integral rangesfrom r : 0 to x : L. Looking up the integralof
sin:(a,rI andsolvingtheaboveequationfor theso-callednormalizationconsranr
I gtves
25
f r
t :
J
llttr )li,/.t: l.-t1:
;l
sin2trzxI L) ttx. 0.72)
A : (2lDtt2
/(-r) : (2111trt sin(nrxI L)
( 1 . 7 3 )
andso
(r.74)
Thevaluesthatn cantakeonaren : l, 2. 3. . . .;thechoiceie: 0 isunaccentable
becauseit wouldproducea wavefunctionry'(.t)thatvanishesat all r.
The full ,r- andt-dependentwavefunctionsarethengivenas
v(,r.r):(:)'rin''l''r.rl-.t'::'!l (r.75) L / L L l t t l L - / l t )
Noticethatthespatialprobabilitydensitylv1,r.r)12isnotdepenclentontimeand
is equalto lry'(.r)12becausethe complexexponentialdisappearswhen v*w is
formed.Thismeansrhattheprobabilityof findingtheparticleat variousvalues
of "r is time-independent.
AnotherthingI wantvouto noticeisthat,unliketheclassicaldynamicscase.
notallenergyvaluesE areallor.ved.IntheNewtoniandynamicssituation,E could
bespecifiedandtheparticie'smomentumatany-r valuewasthendeterminedto
withina sign.In contrast.in quantummechanicsonemustdetermine,by solving
theSchrcidingerequation.rvhatthe allowedl'aluesof 6 are.TheseE valuesare
quantizedmeaningthattheyoccuronlyfordiscretevalues.t : ,2r2h)/(2r,L2)
determinedbv a quantumnumberrr.by themassof theparticlem, andbychar-
acteristicsofthe potential11.in thiscase).
1.3.4Probabilitydensities
Let's now look at sorneof the wave functionsv(.r) and comparethe proba-
bility densitieslv(.r)ir thattheyrepresentto theclassicalprobabilitydensities
discussedearlier.The n : 1 and n : 2 wavefunctionsare shown in the top
of Fig. 1.8.Thecorrespondingprobabilit-vdensitiesareshownbelowthewave
functionsin two tbrmats(asn--vplots andshadedplotsthat could relateto the
flashinglightwayof monitoringtheparticle'slocationthatu'ediscussedearlier).
A morecompletesetof r.vavefunctions(for izrangingfrom 1to 7) areshownin
Fie.1.9.

z o The basicsof ouantummechanics
,ffi'ffin : 2
Thetwo
lowestwave functions
and probability
densities.
Seven
lowestwave functions
andenergies.
{
V.
m,axffita;,;*r
mau,*ffiu,,*;r
Node
25
t 6
9
I
I

Noticethatasthequantumnumbern increases,theenergyE alsoincreases
lquadraticallyrvithn in thiscase)andthenumberof nodesin v alsoincreases.
Also noticethat the probabilitydensitiesare very differentfrom what we en-
counteredearlierfor theclassicalcase.Fo[example.lookatther : I andil : 2
densitiesandcomparethemto theclassicaldensityillustratedin Fig. 1.10.The
classicaldensityis easyto understandbecausewe arefamiliarwith classicaldy-
namics.In thiscase,wesaythatP(.r)isconstantwithintheboxbecausethefact
thattr/(.r) isconstantcausesrhekineticenergyandhencethespeedoftheparticle
to remainconstant.In contrast.ther : I quantumu'avelirnction'sp(.r) plot is
peakedin therniddleof theboxandfallsto zeroat thewalls.Then :2 density
hastwo peaksP(-r)(oneto theleti of theboxmidpoint.andoneto theright).a
nodeat thebox midpoint.andfallsto zeroat thewalls.one thingthatstudents
oftenaskme is "how
doestheparticlegerfrom beingin thelefi peakto beingin
therightpeakif it haszerochanceof everbein_qat themidpointwherethenode
is?"Thedifficr"rltyrviththisquestionis rhatit is posedin a terminologytharasks
foraclassicaldynamicsanswer.Thatis.byasking"horv
doestheparticleget. . ."
oneis dernandingan answerthatinvolvesdescribingits motion(i.e..it moves
fromhereat timet1to thereat time11).Unfortunatell,.quantummechanicsdoes
notdealwithissuessuchasaparticle'strajectory(i.e..rvhereit isatvarioustinres)
butonlyvu'ithitsprobabiltyof beingsomewhere(i.e.,lw l:;. Thenc'xtsectionwill
treatsuchparadoxicalissuesevenfirrther.
1.3.5Classicalandquantumprobabilitydensities
As.lustnoted,it is temptingfor mostbeginningstudentsof quanturnmechanics
to attemptto lnterpretthe qr-rantumbehaviorof a particlein classicalterms.
However.thisadventureis tull of daneerandboundto fail becar-rsesrnall.light
particlessimplydo notmoveaccordingto Nervton'slarvs.To illustrate.lets try
to "undersrand"
r'"'hatkind of (classical)motionwor-rldbe consistenrlvith the
ri : I or n :2 quanturnP(_r)plotsshownin Fig. 1.8.However.asI hope1,ou
anticipate.thisattcrnptatgainingclassicalunderstandingof aquantumresultrvill
not"work"
in thatit r'"'illleadto nonsensicalresults.My point in leadingyouro
attemptsucha classicalunderstandingisto teachyouthatclassicalandquantum
resultsaresimplydifferentandthatyoumustresisttheurgeto imposea classical
understandingon quantumresults.
For the zr: I case,we notethat P(x) is highestat the box midpointand
vanishesat.r : 0 and,r : L. Lna classicalmechanicsworlclthis wouldmean
thatthe particlemovesslowly near-y: L/2 andmorequickly near-r : 0 and
r : L. Becausetheparticle'stotaienergyE mustremainconstantasit moves.
in regionswhere it movesslowlv.the potentialit experiencesmust be high.
and where it movesquickly, tr/ must be small. This analysis(n.b.,basedon
classicalconcepts)wouldleadusto concludethatthen : I p(i arisesfrom tne
27
ClassicaI
probabilitydensityfor
potentialshown.

particlemovingin apotentialthatis highnear.r : L /2 anrlloll,as-r:approaches
0 o r Z .
A similaranalysisof then :2 P(x) plot wouldleadusto concludethatthe
particlefor which this is the correctp(_r)rnustexperiencea potentialthatis
highmidwaybetween:r: 0 andx : L/2,high midu,ayberu,een.r : L 12 and
x = L,andverylownearx: L /2andnear-r: 0andr : r. Theseconclusions
are"crazy"becausen'eknow thatthepotentialt,(-r)for whichwe solvedthe
Schrodin-eerequationrogeneratebothof thewavefunoions(andbothprobability
densities)is constantbetween-r : 0 andt : L. Thatis.."veknowthesamel,(-r)
appliestotheparticlemovingin thell : I andr : 2 states.whereastheclassical
motionanalysisofferedabovesuggeststhat l,(l ) is differentfor thesetwo cases.
what is wron-qrvith our attemptto understandthe quantump(.r) prots?The
mistakewemadewasin attemptingto apprytheequatio'sandconceptsof clas-
sicaldynarnicsto a P(.i-)protthatdid not arisefrom classicarmotion.Siniply
put' onecannot askhow theparticleis mo'ing (i.e..whatis its speedat vari-
ouspositions)whentheparticleisundergoingquantumdynamics.Moststudents.
r,r'henfirstexperiencingquantumrvavefunctionsandquantumprobabilities,trv to
thinkof theparticlemovingin a classicalwaythatisconsistenlwiththequantum
P(-r). Thisattemptto retaina degreeof classicalunderstandingof theparticle's
movementis alwaysrnetwith frustration,asI illustratedu,iththeaboveexamole
andwill illustratelaterin othercases.
continuingwiththisfirstexarnpleof hou'onesolvestheSchrodingerequatlon
andhow onethinksof the quantizedE 'alues andu,avefunctionsv, let me
offera little moreoptimisticnotethanofferedin theprecedingdiscussion.If we
examinethew(.-v)plotshownin Fig.1.9fern :7, andthi'k orthecorresponding
P(;r) : lv(x)12,wenotethatthep(.r-)plotrvouldlooksomethinglikethatshown
in Fig. 1.11.It wouldhavesevenmaximaseparatedby sixnodes.If wewereto prot
I
0.9
0.8
0.7
0.3
0.2
0.1
0
^ . ( A
5
r! 0.5
V o.c
Ouantum
probabilitydensityfor
n: 7 showingseven
peaksand six nodes.
11 iI h
I 1 I
r
II
I
I I
I I
I I
v rJ
xlL

1V (.r)lr for averylarger valuesuchasn : 55,wewouldfinda P(,r) plothaving
55 maximaseparatedby 54 nodes,with the maxima separatedapproximately
b),distancesof(1/55I). Sucha plot,whenviewedin a "coarsegrained"sense
1i.e..focusinuwith somewhatblurredvisionon the positionsand heightsof
the maxima)looksvery much like the classicalP(,r)plot in which P(.r) is
constantfor all -r. In fact,it is a generalresultof quantummechanicsthatthe
quanfumPt.r) distributionsfbr largequantumnumberstakeon the form of the
classicalP(-r)for thesamepotentialZ thatwasusedto solvethe Schrodinger
equation.It is also true that classicaland quantumresultsagreewhen one is
dealingwithheavyparticles.Forexample,agivenparticle-in-a-boxenergvCn:
n:h: I 2ntL2) u,ouldbeachievedfora heavierparticleathighern-valuesthanfor
a lighterpalticle.Hence,heavierparticles,movingwith a givenenergyE. have
higherr andthusmoreclassicalprobabilitydistributions.
!'ervillencounterthisso-calledquantum-classicalcorrespondenceprinciple
again"r,'hen'nveexamineothermodel problems.It is an importantpropertyof
solutionsto the Schrodingerequationbecauseit is whatallowsusto bridgethe
"gap"betrveenusingtheSchrodingerequationto trearsmall,lightparticlesand
theNewtonequationsfor macroscopic(big,heavy)systems.
Anotherthing I u'ouldlike you to be arvareof concerningthe solutionsry'
and.Eto this Schrodingerequationis thateachpair of wavcfunctionsry',,and
r/r,,belongingto difl-erentqltantumnumbersn andn'(and to differentener_qiesl
displaya propert,vterrnedorthonormality.Thispropertymeansthatnotonlyare
tfr,,and,{r,, eachnormalized
29
r t
l : l t r l t , , : , / t : / . a , , : l t ' .
J , J
buttheyarealsoorthogonalto eachother
o=
lr',t,,,r',t,,,a,
( 1 . 1 6 )
( 1 . 7 7 )
wherethe complexconjugate* of the first function appearsonly when the ry'
solutionscontainimaginarycomponents(youhaveonlyseenonesuchcasethus
far- theexp(ill @) eigentunctionsof the:-componentof an-qularmomentum).It
iscommonto writetheintegralsdisplayingthenormalizationandorthogonality
conditionsin thefbllorvineso-calledDiracnotation:
| - / r lr , ,
| 11, , 0: 1r lt , , 'r ! , , ) . ( 1 . 7 8 )
wherethe l) and (l si'mbolsrepresenttb andlt*, respectively,and puttingthe
two togetherin the (l) constructimpliestheintegrationoverthevariablethatry'
dependsupon.
Theorthogonalityconditioncanbeviewedassimilarto theconditionof two
vectorsv1 and v2 beingperpendicular.in which casetheir scalar(sometirnes
called"dot")productvanishesVl . vr : 0.I wantyouto keepthispropertyin mind

5U Thebasicsof quantummechanics
becauseyour.l'illsoonseethatit is a characteristicnotonlyof theseparticle-in-
a-boxwavefunctionsbut of all w,ar.,efunctionsobtainedfrom any Schrodinger
equatlon.
In fact,the orthogonalitypropertyis evenbroaderthanthe abovediscusston
suggests.It turnsoutthatall quantummechanicaloperatorsforrnedasdiscussed
earlier(replacingCartesianmomentap by thecorresponding-ihd ldq operator
andleat'ingall Cartesiancoordinatesasthevare)canbe shor'vnto be so-called
Hermitianoperators.This rneansthattheyform Hermitianmatricesu'henthey
areplacedbetweenpairs of functionsand the coordinatesare integratedover.
Forexample,the rnatrixrepresentationof an operatorF vu'henactingon a setof
functionsdenoted{dr} is
FrL: (QttFtQLt:
|
*;rr' u, (1 . 7 e )
For all of the operatorsformedfollowing the rulesstatedearlier.onefindsthat
thesematriceshavethefollowingproperty:
r _ r +r J I - r / t . r1 . 8 0 )
rvhichmakesthematricesu'hatwe call Hermitian.lf tlrefunctionsuponu'hich
F actsandF itseifhaveno imaginaryparts(i.e..arereal).thenthetnatricesturn
outto be svmmetric:
Fr.L: Ft.r (r . 8 l)
Theimportanceof theHermiticity oI symlnetryof thesematriceslies in thefact
tl.ratit canbeshownthatsuchmatriceshaveallreal(i.e.,notcor.t.rpler)eigenvalues
andhaveeigenvectorsthatareorthogonal.
So, all quantummechanicaloperators,not just the Hamiltonian,havereal
eigenvalues(this is good sincetheseeigenvaluesarervhatcanbe measuredin
any experimentalobservationof thatproperty)andortho-qonaleigenfunctions.
It is importantto keepthesefactsin mind becausewe makeuseof them many
timesthroughoutthis text.
1.3.6Timepropagationof wavefunctions
Fora systemthatexistsin aneigenstateV(x) : (2lL)t12 sn(nitx lL) havingan
energyEn : n2r2h2lQmLz),the time-dependentu'avefunctionis
v(x.r): (?)"',inff*r(-?), (1 .8 2 )
which can be generatedby applying the so-called time evolution operator U(l' 0)
to the wave function at I : 0:
V(x.r): U(r,0)v(,r,0) (1.83)

( 1 . 8 7 )
( r . 8 8 )
rvhereanexplicitform for U(t, t') is
L(r.r':.*o[-"' *'''"-l. (1.8+.)
L n l
Thc-functionV(.r, t) hasa spatialprobabilitydensitythatdoesnot dependon
timebecause
v.(.r.r)v(.r.,,: (;) ""' (?) (1 . 8 5 )
sinceexp(-rtE,'l/i) exp(itE,,l/i) : l. However it is possibleto preparesystems
(evenin reallaboratorysettings)in statesthatarenot singleeigenstates;we call
suchstatessuperpositionstates.Forexample,considera particlemovingalong
the-r--axiswithin the"box" potentialbut in a statewhosewavefunctionat some
i n i t i a l t i m e / : 0 i s
, / l l : / 1 7 r . . r 2 t l : / l - z . r '
u ( r o r : , , , ( ; ) ' ' ' (
. ) - , , ( ; ) , ' " ( ? )
{ r 8 6 )
Thisisasr-rperpositionofthen:Iandz:2eigenstates.Theprobabilitvdensity
associatedn iththisfunctionis
r,r,r': I |;,'",(?) +/ sin:(?) - r(?)
/ lrr /1.-r.rt I
' t ' n (
r / t ' " ( . , / i
Then : I andn : 2 components,thesuperpositionV. andtheprobabilityden-
sityat1 : 0lV lr areshownin thefirstthreepanelsof Fig.l. 12. It shouldbenoted
thattheprobabilitydensityassociaredwith thissuperpositionsrateis not sym-
metricaboutthe"r: I/2 rnidpointeventhoughthen : I andn : 2 component
wavefunctionsanddensitiesare.Sucha densitydescribestheparticlelocalized
morestronglyin thelarge-"vregionof thebox thanin thesmall-,rregion.
Now. let'sconsiderthe superpositionwavefunction and its densitl,at later
times.Applying the time evolutionoperatorexp(-itHl/i) to V(r,0) generates
thistime-evolvedtunctionattime/:
v(;rr1:*'(-#)
{,
'{i) ',,"(T)-''
:
['
'(;)"''(?)'-o(-i)-''
"',"(?).-'(-+)]
Thespatialprobabilitydensityassociatedwith this W is
'(?r)",'"
(?)l
,'G)"'
rv(r,)r::j{(;),'",(+). (;),,'',(?)
-'(?).o,
[ru,
-u,)if,',(+),'"(?)] o8e)

5Z
r I--r
/ i I
/ f
lt
X , L
--
! _ t
t I
I
/ l
t
t
I
j
! 7
f_'
.l
I
{
t
=
I
d o.E
E t,.o
l i . . -
.-
^ _0.1
{
> { ) l
=
- t
{
6l
I
vi
u.l
0 ^
0 6
0.i
0 4
(1.-l
( ) l
0
(1.I
- 0 :
1 . 6
t . l
J
0 N
{
t:
I
.
t:
.=
:
0.6
1).1
At r : 0' thisfunctionclearlyreducesto thatu,rittenearlierfor v(r.0). Notice
thatastimeevorves.thisdensitychangesbecauseof thecos[(82- E 1ltf til factor
it contains'In particular,note that asr movesthrougha period of rength6l :
n h/ (E2 - f r).thecosfactorcha'gessign.Thatis.for r : 0.thecosfactoris + l ;forl : r/i/(E2 - tr), thecosfactoris_l;for t :2r/il(Ez_ Er ).itreturnsto
* L Theresultof thistimevariationin thecosfactoris thatlV l: .lrun-c.,in forn.r
from thatshownin thebottomleft panerof Fig. l .12to tr-ratshou,nin thebottom
right panel(at t : trril(E2 - 6r)) andthenbackto the fbr'i in the bottomreftpanel(att : 2tr11/(Ez- E t)).one caninterpretthistimevariationasdescribing
theparticlekprobabilitydensity(not its classicarpositionl),initiailyIocalized
towardthe right sideof the box, movingto the left and thenback to the rr_qht.
of course,thistimeevolutionwill continueovermoreandmorecycresastrnre
evolves.
This exampleiilustratesonceagainthedifficulty with attemptingto locarize
particlestliatarebeingdescribedby quantumwave-functions.Forexampre.apar-ticlethatis characterizedby theeigenstate(2/D) /2 sin(tr x I L) i, rn"rill*.,u ,"bedetectedfleor'i : L/2thannearir :0 orrj - r becausethesquareof thisfunctionislargenear-r= L/2.Aparticleinthestate(2lDt/2rin1:n..7tj,smort
likeliztobefoundr€&rx: L/4 andx : 3Lf4,butnotnear,r: 0,x : 11t.sy
^l i.
/
f
1
i
I
d a
|I,
-1^
I
I

I
a
)
L_] -
jflff rhe n: 1 and n: 2 wavefunctions,theirsuperposition,andtheandtime-evolvedprobabilitydensitiesof thesuperposition.

.r = L. The issueof horvthe particlein the latterstatemovesfrom beingnear
r = L f 4 tox : 3L l4 isnotsomethingquantummechanicsdealsrvith.euanfum
mechanicsdoesnot allow usto follow theparticle'strajectory.which is whatwe
needto knowwhenweaskhow it movesfrom oneplaceto another.Nevertheless.
superpositionwavefunctionscanoffer"to someextent,theopportunityto tbllow
themotionof theparticle.Forexample,the superpositionstatewrittenaboveas
2-r,:12111112sin(lz-rlZ)-2-t/2QlL)t/2sin(2nxlL)hasaprobabilityampli-
tudethatchangeswith time asshownin the figure.Moreover,this amplitude's
majorpeakdoesmovefrom sideto sidewithin the box astime evolves.So,in
thiscase.weciln saywith whatfrequencythemajorpeakmovesbackandforth.
In a sense.this allor.vsus to "follorv"
the particle'smovements.but only to the
extentthat'"vearesatisfiedwith ascribingitslocationto thepositionof themajor
peakin itsprobabilitydistribution.Thatis,we cannotreallyfoilor.vits "precise"
location.but we can follor,vthe locationof whereit is very likely to be found.
ThisisanimportantobservationthatI hopethestudentwill keepfreshin mind.It
is alsoanimportantingredientin modernquantumdynamicsin which localized
wavepackets,similarto superposedeigenstates,areusedto detailtheposition
andspeedof a particle'smainprobabilitydensitypeak.
Thc-aboveexampleillustrateshowonetime-evolvesa ryavefunctiontharcan
be expressedasa linearcombination(i.e..superposition)of eigensratesof the
problemat hand. There is a large amount of current eftbrt in the theoreti-
cal chemistrycommunityaimedat developingelicient approximationsto the
exp(-i t H lll)evolutionoperatorthatdo notrequireV(.r.0) to beexplicitl_yrvrit-
tenasa sumof eigenstates.Thisis importantbecause,formostsystemsof direct
relevanceto molecules,onecannot solvefor the eigenstates;it is simplytoo
difficult to do so. You can find a significantlymore detailedtreatmentof this
subjectattheresearch-levelin my TheoryPagewebsiteandmy el,tIC tertbook.
However.let'sspenda little time on a brief introductionto whatis involved.
The problemis ro expressexp(-itHlh)V(qr), whereV(qr) is someinitial
wave function but not an eigenstate,in a mannerthat does not requireone
to first find the eigenstates{V7} of H and to expandV in terms of these
eigenstates:
v : f c , w ,
J
afterwhichthedesiredftlnctionis writtenas
(1.e0)
J5
( l . e r )
Thebasicideaisto breakH intoitskinetic I andpotentialtr/energycomponents
andto realizethat the differentialoperatorsappearin I only.The importance
of this observationlies in the fact that I and Z do not commute,which means
thaLTV is not equalto ZI (n.b.,for two quantitiesto commutemeansthattheir
"t(-#)v@,):
1,,*,*,(-+)

orderof appearancedoesnot t.natter).Why do they not conrmute'lBecauseI
containssecondderivativesn'ithrespecttothecoordinates{q, }thatI' dependson,
so.for example,d2I dq2LI,@)V(q )l is notequalto Il(q) d2i d(12V (r7).Thefact
that r and l' do not commuteis inrportantbecausethe rnostcommonapproach
to approxrmatingexp(-rlH/fi) is to rvritethis singleexpor.rentialin rermsof
exp(-itTlh) andexp(-lri.'/fi). However.theidentity
*'(-f):*'(-+)*'(-f) ( l q 2 )
is not fully validasonecanseeby expandingall threeofthe aboveexponential
factorsasexp(.r): I *.r- + x2/2! *.... andnorin_qthalthe r$,osidesof thc
aboveequationonlya-ereeif onecanassulnethatTl': I'L u,hich.asrvenoted-
is nottrue.
In mostr.nodernapproachesto timepropagation.onedividesthetimeinterval
r intonrany(i.e..P of them)smalltime"slices"
r : t/ p. onc thenexpresses
theevolutionoperatorasa productof P short-timepropagators:
/ i t H / r r H l I i t H r i r H t
e x p ( - , ,
/
: . * o l - - ) . - o ( - ; , ) . - n ( -
; )
I t i r t l l r
:
L'*n-t /l ( r . e 3 )
If one can developan efficientrneansof propagatingfor a short time r. one
canthendo so over and over againP timesto achievethe desiredfull-time
propagation.
It canbe shownthatthe exponentialoperatorinvoll'ing 11canbetterbe ap-
pro.ximatedin termsof the f and I/ exponentialoperatorsasfollows:
/ r r H l . t f t ' - V n / i r r ' r i r r e x p -
f t / - . * p t , - .
- - - F - l . * p ( -
a / . - o ( _ ; /
( r . e _ r )
So.if onecanbe satisfiedwith propagatingfor very shorttirneir.rtervals(sothat
therl termcanbe neglected),onecanindeeduse
*'(-#)-*p(-+)-'(-T) (r.e5)
asanapproximationfor thepropagatorU(r, 0).
To progressfurther,one then expressesexp(-irT//1) acringon the initial
function v(q) in terms of the eigenfunctionsof the kinetic energyoperator
r. Note that theseeigenfunctionsdo not dependon the natureof the poten-
tial z, sothis stepis valid for anyandall potentials.Theei_eenfunctionsof r :
* 1 , ^ r t , , f
-n-
/ tntd'/ctq' arc
,!,,(q):(*)'"*r(T) (1.e6)

andtheyobeythe following orthogonality
35
f
J
Vi,tattlr,,(qtdq: 3rp - pl
andcompletenessrelations
J
rL,.t,ttrlr,",tt1lp = |rq - q t.
WritingV(q) as
v(q): I Au- (]')v(q')dq,.
andusingtheaboveexpressionfor d(q - q') gives
rr(q) :
| |
v,,u,,,t';,,q')rtt(q'ytq'ttp.
(r.97)
(r.e8)
(1.e9)
( 1 . 1 0 0 )
Tlreninsertin-etheexplicitexpressionsfortlr,(c1)andr!;(q' ) in termsof ,,ft,,1c11:
0 l2ir 1tr'zexp(ipr7/fi) -erves
. f f l I t r : 1 i 7 r 4 1 7| 1 r : / i p utQ t ' 7 t :
J I ; J
' * o ( ; l ( ; , ) . . - o ( - # ) Q r ' 1t ' t ' 1 ' t pr r r 0 r t
Now.allor.vingexp(-i r TlIt)to acton thisform for W(r7)produces
/ i r T r r / ; ' , ' r t i / I ' '. . P (
n ) w t q , :JJ . ' p ( ' # l l r ;t
^.*o[?,u:,,
,l
f :)''' *,r,ror,r, (rr02)' L
h ) h t
Theintegraloverp abovecanbecarriedout analyticallyanclgives
/ i r T . 1 m r : / f i n r q _ 4 ' t : l.*P( n )q,,t,: (:r,r,)
7.*oL ," 1
ttt,t'),t"t. rr.l03r
So,the final expressiontbr the short-timepropagatedwave function is
v { , r . r ) : . * o [ - i r l i ' l
t f , n t ' t f - . ^ f i n t 1 t 1- ( t 7 : 1
" ' , " - " u L
/ r - j ( . , , r /
7 . * p L , * l V t t 1 ' t d , l .
{ 1 . 1 0 + )
whichistheworkingequationoneusestocomputeV (q, t )knowingV (4).Notice
thatall oneneedsto kno."r,to applythisformulaisthepotentia1v (q) ateachpornt
ln space.one doesnotneedto knowanyof theeigenfunctionsof theHamiltonian
to applythis method.Howe'u'er.onedoeshaveto usethis formulaoverandover
agarnto propagatethe initial wavefunctionthroughmanysmalltime stepsr to
achievefr"rllpropagationfor the desiredtime intervalt : pt.
Becausethistypeoftime propagationtechniqueisaveryactiveareaofresearch
inthetheorvcommunity.it islikelytocontinuetoberefinedandimproved.Further
discussionof it is beyondthe scopeof this book,so I will not go furtherin this
direction.

1.4 Freeparticlemotionsin more dimensions
Thenumberof dimensionsdependson the numberof particresand the
number of spatial {and other) dimensionsneededto characterizethe
positionand motion of eachparticle.
1.4.1TheSchrodingerequation
consideran electronof massri and chargee mo'rng on a tlvo-dimensionar
surfacethatdefinesthex, -r.plane(e.g.,perhapsanelectronis constrainedto the
surfaceof a solid by a potentiarthat bindsit tightly ro a narro,regio' in the
---directio.).andassumethattheelectronexperiencesa constantandnot trrne-
varvrngpotentialI1tat allpointsin thisplane.Thepertinenttinre-independent
Schrodingercquationis
t / a t a t . .-t
a"t'
+ 'drt
)t(r't')*
Litlt(x"r'):Ef('r"r') (1 10-5)
The taskat handis to solvethe aboveeigen'alueequationto deternrinethe"allorved"
energystatesfor thiselectron.Becausetherearenotermsin thisequa_
tionthatcouplemotionin the; andl directions(e.g..noternlsof theform,t,-r.b
or Sldx 0/Ev orx 0/8v),separationof'u'ariabrescanbeusedrowritey'rasaprod-
uctU(r. -r'): l(.r)BLr,).Substitutionof thisforrnintotheSchriidingerequarion,
followedby collectingtogetherallr-dependentandall_r'-dependentterms,_qrves
r f 1 a ] A r t t a 2 a
2 * 7 a r ; - 2 - E * : E - r t t '
( l ' 1 0 6 )
Since the first term containsno r,-dependenceand the secondcontainsno
x-dependence.andbecausetherightsideofthe equationis independentofboth
x and-r,,both termson the left must actuallybe constant(thesetwo consranrs
are denotedE" and Er', respectiveiy).This observationailows two seDarare
Schrcidingerequationsto be written:
rt . alA- - / - ' - - F -
ztn dx.
( 1 . 1 0 7 )
and
il ^-, fB-
2m
b
ai
= r)'' (l'lo8)
ThetotalenergyE canthenbeexpressedin termsoftheseseparateenergiesE,
andE, asE, + E:, : E - 20.Solutionsto thex- and,t,-Schrcidinge..quations
areeasilyseento be:
A(x):*o[,,.(+)"']
B(,r'):.*p[,r,(ry)"]
I r ' - l
. | / 2 n E .t ' - |
a n o e x p l - i l t - . 1 | ( l . t 0 9 t
L t r / l
and .^o[-,,(+)"1 (rro)
L r )

Freeparticlemotionsin moredimensions
Trio independentsolutionsareobtainedfor eachequationbecausethe,r- and
.r.-spaceSchrodingerequationsarebothsecondorderdifferentialequations(i.e..
a secondorderdifferentialequationhastwo independentsolutions).
1,4.2Boundaryconditions
The boundaryconditions,not the Schrodingerequation,determine
whetherthe eigenvalueswill be discreteor continuous.
If theelectronisentirelyunconstrainedwithinthe-r._r'plane,theenergiesE,
and8.,.canassumeany values:this meansthatthe experimentercan"inject"
rheelectronontothe,r.r'plane with anytotalenergyE andanycomponents
8., and^t.,.alongthetwo axesaslongasE^ + E;. : E .In sucha situation.one
speaksoftheenergiesalongbothcoordinatesasbeing"in thecontinuuln"or'.not
quantized".
Incontrast.if theelectronisconstrainedtoremainwithinafixedareainthe.r.,r,
plane(e.g.,arectan-eularor circr.rlarregion),thenthesituationisqualitativelydif'-
i'erent.constrainin-etheelectronto anysuchspecifiedareagivesriseto boundary
conditionsthatimposeadditionalrequirementson theaboveA anJB fLrnctrons.
Theseconstraintscanarise,for example,if thepotentiall,ii(.r._r,)becornesverv
Iargetbr,r..i valuesoulsidetheregion,inr,vhichcaserheprobabilityof fincjinethe
electronoutsidetheregionisverysmall.Suchacasemightrepresent.forexample.
asituationinrvhichthentolecularstructurcofthcsolidsurfacechangesoutsidcthe
enclosedregionin a'aythatishi-uhlyrepulsiveto rheelectron(e.g..asin thecase
of molecularcorralson metalstrrfaces).Thiscasecouldthenrepresenta simple
modelof so-called"corrals"
in rvhichtheparticleisconstrainedtoa finiteresion
of space.
Forexarnple.if rnotionisconstrainedto takeplacelvithina rectaneularregron
definedby 0. -r < l,:0 <,r.< 1.,.thenthecontinuitypropertythatall wave
tunctionsrnustobev (becauseof their interpretationas probabilitydensities.
whichmustbecontinuous)causesI ("r) to vanishat0 andat r , . Thatis.because
.'1mustr.'anishfbr.r < 0andmustl'anishfor.v> z.r,andbecauseA iscontinuous.
i t m u s t v a n i s h a t , r : 0 a n d a t . r : 2 . , . L i k e r . r , i s e . B ( r r ) m u s t v a n i s h a t 0 a n d a t Z . . .
Toinrplemc'nttheseconstraintsfor I (.r:).onemustlinearlycombinetheabovetwo
solutionsexp[i,r(2rrE,lli2ltizlandexp[-r.r(2nE,y/t2lrizltoachieveiifunction
thatvanishesat,r : 0:
'71
( r . l ll )
one is allowedto linearlycombinesolutionsof the Schrodingerequationthat
havethesameenergy(i.e.,arede,eenerate)becauseSchrcidingerequationsare
r(r1:e1p
l',(rf)'"] -.*o
[-,,{i+)
"]

J d
li,ear differentiarequations'An analogousprocessmustbe appriedto g(,r,)toachievea functionthatvanishesat.r,: 0.
B(r-opf,.',(t+!"'l -.*"f ..12,,r,1 :.1
' L t i I l -
q r ) L - / 1( ; /
J
r n r 2
Furtherrequirin-eA(.r) andB(rr)to,,,anish.respecti'elrrat.r : Z. and_r,
_ 2,.sriesequationstharcanbeobeyedonlyif ,f, andE, assunreparticularvalues:
T
. r n 1 , , , / 2 n E t t : f |
, i a r E , 1 i : l
L ( l - ) l -
e r n J - i r '( ; ) l = ' r ir . , r
and
f , 2 n t E i ,
, l
[ / ] 1 1 1 6 ,r ,
r l
e r Pj ir . ( f ) I
' ^ n l - ' r '
; - ) l : , r r r 4 rL " , J L l
Theseequationsareequivalent(i.e..usingexp(i.r): cos-r,* i sin_i,)to
. , - [ , 1 2 n t E ; l : l l - r l a i E ,l r : ls ' n l r ( - - F - )
l = . i n l z(
L J L ' r i / J : u
r r ' i r 5 t
Knou,ingtharsing vanishesate- nn,forz : l. 2. 3. . . . lalthoughthesin(zr)function'u'anishesforn :0, thisfunction'anishesfor all-t or.r,,andisthereforeunacceptabrebecauseit representszeroprobabiritydensityatalrpointsin space).oneconcludesthattheenergiesE, a'd.E, canassumeo'ly'alues thatobey
t 1 , , , t r , l / 2
z , { 1 + r I = r , o .
/ r /
. 1 2 n E , r ' 2
r ' I ' . - l : n , r ,
f l /
^r , - nlr2rt
zilr Li
nln2tlt n d k -dffu t, =
T;8.
uith n and
( t . l l 6 )
( 1 . t 1 7 )
( l . r l 8 )
t l ,= 1 . 2 . 3 , . . . ( l . l1 9 )
It is importantto stressthat it is the impositionof boundaryconditions,ex-pressingthe fact that the electronis spatia'y constrainedthat gives ,se toquantizedenergies.In the absenceof spatialconfinement"or wjth confinementonly at;r : 0 or Z, or only at y _ 0 or 2.,, quantizedenergieswould rot berealized.
In this exampre,confinementof the erectronto a finite intervalalong boththe 'r and'r"coordinatesyields energiesthat are quantizedalong both axes.Iftheelectronis confined"t:lg:l".oJrdinur.
1.g., u.t*".n 0 < x < 2..) burnotalongtheother(i.e.,Blu) is eitherrestricteaioi,anist aty - 0 or at .1,: L,. oratneirherpoinl).thenthetoralenergyE liesin th,
isquantizldu,ir, i, ""t.Anarogsorsuch;:Tl'.::T;:li.*r;:,TiffTiineartriatomicmorecurehusmo.i thanenouj energyin one of its bondsto

ruprureit but not muchener,{yin the otherbond;the first bond,sener-evlies in
thecontinuum.br"rtthe secondbond'senergyis quantized.
Perhapsmoreinterestingisthecasein whichthebondwiththehigherdissoci_
ationenergvisercitedto a levelthatisnotenoughto breakit butthatisin excess
ofthedissociationenergyofthe weakerbond.In thiscase.onehastw.odegenerate
states:(i) thestron-sbondhavinghi-ehinternalener-qyandtheweakbondhaving
low energy(/r). and(ii) thestron_ebondhavinglittleenergyandtheweakbond
ha'ing morethanenoughenergyto ruptureit (tL). Althou_ehanexperimenrmay
preparethe moleculein a statethatcontainsonly the formercomponent(1.e.,
rlt : Crltr -f C1r2.with cr - l. c: : 0), coupringbetrveenthetwo degenerate
functions(inducedby termsin the HamiltonianH that havebeenignoredin
definingry'1andry'2)usuallycausesthetruewavefunction W : exp(_il Hih){.,
toacquireacomponentof thesecondfunctionastimeevolves.In suchacase,one
speaksof internalvibrationalenergyrelaxation(IVR)givingriseto unimolecular
decompositionof thenrolecule.
1.4.3Energiesandwavefunctionsfor boundstates
For discreteenergyrevers,the energiesare specifiedfunctionsthat de-
pend on quantum numbers, one for each degree of freedom that is
quantized.
Returningto thesituarioni'which motionisconstrainedalongbothares.the
resultanttotalenergiesandwavefunctions(obtainedby insertingthequantum
ener_cylevelsinto theexpressionsfor '1(,r-)g( r.))areasfollor,",s:
- ,l:iT- tT
'
2 n L i
n:n-lr
t r _L ' - :
- .
tnt Ll.
E : E , + 8 , + t r / o .
(r.120)
(r .r 2 l )
1 . t 2 2 )
?a
1 | : / I T , : , ,
y'(,r :(+) (#)'L*o("F)-.*o(#)]
.[.-o(+)-.-,(-i:)]
w i t hn , a n di l , = ! . 2 , 3 , . . . ( L . l : l )
Thetwo (1/2Dt/2 factorsareincrudedto guaranteethatry'is normalized:
(L 12,{)
NormalizationallowsltL(x, y)r2to beproperlyidentifiedasaprobabilirydensity
for findingtheelectronat a point-lr.r,.
t
wo.1,)l2dxttv: I

40 The basicsof quant um m ech a n ics
Shou'nin Fig. l.l3 areplotsof four suchtwo-dimensionalu,avefunctions
for rr. andn, valuesof (1,1),(2,1),(1.2)and(2.2).respecti'cl.v.Norethatthe
functionsvanishon the boundariesof the box,andnoticehou,tlrenurnberof
nodes(i.e..zeroesencounteredasther.r,avefunctionoscillatesfi-ompositiveto
ne-qati:e)is relatedto the r- andn.,.quantun nurnbersandto the energy.This
patternof morenodessignifyinghigherener_rlyis onethatu,eencounteragain
andagainin quantummechanicsandis somethingthe studentshouldbe able
to useto "guess"
the relativeenergiesof war.efur.rctionswhentheirplots areat
hand.Finally,you shouldalsonoticethat.asin the one-dimensionalbox case.
an)'attemptto classicallyinterprettheprobabilitiesp(-r.
-r,)correspondingto the
abovequanturn4'avefunctionswill resultin failure.As in theone-dinrensionai
case.theclassicalP(.:r'.1,)wouldbeconstantalongslicesof fixed.r-andvaryinu_i,
orslicesoffired,r'andvarying-rr.r,ithintheboxbecausethespeedisconstantther
Hou'ever,thequar.rtumP(x, t,)plots,atleastfor smallquantumnumbers,arenot
constant.Forlarger, andn,.values,thequantump(r,
l,) plotswill again,viathe
quanturn-classicalcorrespondenceprinciple.approachthe (constant)classical
P(.r.,i,)form.
Plots of the
( a )( 1 , 1 ) ,( b )( 2 , 1 ) ,( c )
{1,2)anddl Q,2lwave
functions.
rffi)
wl

1.4.4Quantizedactioncanalsobe usedto derive
energylevels
Thereis anotherapproachthatcanbeusedto find energylevelsandis especially
straightforwardto usefor systemswhoseSchrodingerequationsare separable.
Theso-calledclassicalaction(denoteds) of aparticremovingwithmomentump
alongapathleadingfrom initialcoordinateq1atinitialtimeri to afinalcoordinate
qr.artimer; isdefinedby
s =
/*"'R.o{.
4 1
( r . 1 2 5 )
Here.themomentufirvectorp containsthemomentaalongall coordinatesof the
system.andthe coordinatevectorq likewisecontainsthe coordinatesalongall
suchdegreesof freedom.Forexample,in thetwo-dimensionalparticle-in-a-box
problemconsideredabove,q : ("r..ir)hastwocomponentsasdoesp : (p,.p,,),
andtheactionintegralis
( r . r 2 6 )
Incomputin-usuchactions,it isessentialtokeepin mindthesignof themomentum
astheparticlemovesfrom its initialto itsfinalpositions.An examplewill help
clarify'thesematters.
Forsystemssuchastheaboveparticle-in-a-boxexamplefor whichtheHamil-
tonianis separable,theactionintegraldecomposesintoa sumofsuch integrals.
onefbr eachdegreeof freedom.In this trvo-dimensionalexample,theadditivity
of H.
H : t i ,- H ,: + - + + v t y t 1v l v t
2n 2n
ti ;;t: fi a2
; :
- r I ' { . t ) - : - : - ; * l ' ( r ' ) .
tttt tt.Y- llll dl'-
meansthatp. andp.r.canbeindependentlysolvedfor in termsof thepotentials
v(r) andI,'(,u)aswell astheenergies6. and6,, associatedwith eachseparate
degreeof freedom:
p , : + r / 2 i ( . 1 r l o ) ) ,
, / = -
P t = t t t l b - , - Y l . v l l ;
the signson p, and pr, must be chosento properlyreflectthe motion that the
particleis actuallyundergoing.Substitutingtheseexpressionsinto the action
rntegralyields
s : s , + , s , ( 1 . 1 3 ( ) )
( r . 1 2 7 )
(r.r28)
( 1 . 1 2 9 )
f
fi:rf
f.tfxt
:
J,,,,,
t f2n1E,- v(t))rr *
Jr,,,,
t Jrm@,- vtrt))t/.v.(r.l-ll)

The relationshipbetweentheseclassicalactionintegralsandthe existenceol
quantizedenergylevelshasbeenshownto involveequatingtheclassicalaction
for rnotionon a closedpathto anintegralmultipleof planck'scorlstant:
s.rn."d= fqt=c' :ttp.dq: rilr (, : l. ].3.:1....). (1.132)
Appliedto eachof theindependentcoordinatesof thetrvo-dirnensionalparticle-
in-a-boxproblem,thisexpressionreads
l , = L r - ' r
n. , lt=
| , 2t n( E, - I'(,r))d t + |
- !f/x(l' Jr(.rD./r. (l ljj)
J.-o -l,=r.
f , = t r f , t
n , h : I t t 2 m ( E ,- 1 " ( . r . ) )t l r + |
* r , 1 i 1 E _ t 1 r y 1, l r . i l . 1 3 . 1 )J r : 0 " l t : t ,
Noticethat the signsof the niomentaare positivein eachof the first inre-
gralsappearingabove(becausetheparticleis mo'ing fi.om.r: ()to.r-:1,.
andanalosouslyfor.t-lttotion.andthushasposirirenrontcntullrrnd nelarirt
in eachof the secondinregrals(becausethernotionis from.r: l, to -i-:0
(andanalogouslyfor.r'-motion)andthustheparticlehasnegativenromentum).
Withintheregionboundedby 0 . -r-< l.:0 <,r. < 2,. thcpotentlalvanishes"
so i/(x) : l'lr') : 0. Usingthis fact.andre'ersingtheupperantilou,erlinrits.
andthusthesi-sn.in thesecondintegralsabove.oneobtains
n , h : 2 [ ' = t ' , ' 7 , , , qd r : 2 r 5 u , P 1 , .
n, h : 2
[
'='
1Zn t, tlt.: 2{r,, tr, L,
Solving for E, and 8,. one finds
- ( n , h 1 2
6ri l-;
- ( n ' h 2
snLl
Thesearethe sarnequantizedenergylevelsthatarosewhenthe u,avefunctron
boundaryconditionswerematchedatx : 0.r : 2., and,l : 0..1,: 1,,.In this
case!onesaysthattheBohr*sommerfeldquantizationcondition.
nh:
lo't=o""
p.de,
( l . l i 5 )
( 1 . 13 6 )
( r . r 3 7 )
( 1 . 1 3 8 )
( 1 . r 3 e )
hasbeenusedto obtaintheresult.
Theuseof actionquantizationasillustratedabovehasbecomeaveryimportant
tool. It has allowedscientiststo makegreatprogresstowardbridging the gap
betweenclassicalandquantumdescriptionsof moleculardynamics.In particular,

byusingclassicalconceptssuchastrajectoriesandthenappendingquantalaction
conditions.peoplehavebeenableto developso-calledsemi-classicalmodelsof
moleculardynarnics.In suchmodels,oneisableto retainagreatdealof classical
understandingwhiie building in quantumeffectssuchas energyquantization.
zero-pointenergies.andinterf'erences.
1.4.5Ouantizedactiondoesnot alwayswork
Unfortunately.the approachof quantizingthe actiondoesnot alwaysyield the
correcterpressionfor the quantizedenergies.For example.when appliedto
theso-calledharmonicoscillatorproblemthatwe will studyin quantumform
later.whichservesasthesimplestreasonablemodelfbr vibrationof a diatomic
moleculeAB. oneexpressesthetotalenergyas
e : { + , ' ( l . l . r o )
tlt t
where4: m^ntBl0nA f rrs) isthereducedmassoftheAB diatom.fristheforce
constantdescribingthebondbetweenA andB.r isthebond-lengthdisplacement.
andp isthemomentumalongthebondlen-eth.Thequantizedactionrequirement
thenreads
( l . l . 1 l )
Thisinte-uraliscarriedoutbetweeny = -(28lk)l'l andCE lkl
'1.theleftand
rightturningpointsof theoscillatorymotion.andbackagaintoformaclosedpath.
Carryingout this integralandequatingitto nh givesthe followin_qexpression
forthc energyE:
( 1 . 1 4 2 )
rvheretheqLlantumnunrberri is allorvedto assumeintegervaluesrangingfrom
n : 0. l. 2.to infinitv.Theproblemrviththisresultisthatit iswrong!Asexperi-
mentaldataclearlyshor'",.thelowestenergylevelsfor thevibrationsof arnolecule
do nothaveE :0: thevhavea "zero-point"energythatis approrimatelyequal
LoIl2(hl21 11.1;I pti: . So.althoughtheactionquantizationconditionyieldsen-
ergieswhosespacingsarereasonablvin agreementrvithlaboratorydatatbl lorv-
energystates(e.g..suchstateshaveapproxirnatelyconstantspacings),it failsto
predictthezero-pointenergycontentof suchvibrations.As we will seelirter,a
properquantumrrechanicaltreatmentof theharmonicoscillatoryieldsenc'rgies
of theibrm
43
,h: I pd,: I l',('-1,')]'',,
, , . 1 2
I I I K
6 - : , , . l - l: : r l L /
E : ( , , . : ) ( * ) ( : )
"
( 1 . 1 4 3 )
which differs from the oction-bascdresultby the proper zero-pointenergy.

Evenu'ithsuchdifficultiesknou'n.nruchprogresshasbeenmadein extending
themostelementaryaction-basedmethodstomoreandmoresvstemsbyintroduc_
ing.for exanrple.rulesthatalloll thequantumnumbern to assumehalf-integer
as'"vellasinte-qervalues.Clearl1,.i1'nu'ereallowedto equall/2.3/2.512....,
theearlieractionintegralwouldhaveproducedthecorrectresult.However.hou,
doesoneknow whento allorvr?to assumeonly integeror only harf-integeror
bothintegerandhalf-integervalues?Theansr.versto thisquestionarebeyondthe
scopeofthis textandconstituteanactiveareaofresearch.Forno$,.it is enough
for thestudentto beawarethatonecanoftenfind energylevelsby,usingaction
integrals.butonerrrustbecarefulin doingsobecausesolnetlmestheanswersare
wrong.
Beforeleavingthis section.it is worth notingthat the appearanceof half-
rntegerquantumnumbersdoesnotonlvoccurin theharmonicoscillatorcase.
-lb
illustrate.let us considerthe r_-angularmomentumoperatordiscussedearlier.
As we showedthis operator.whencon.rputedasthe---componentof r x p, can
bewrittenin polar(r.0.6) coordinatesas
L_ : -ihd /d0. ( l . l 4 l )
Theeigenfunctionsofthis operatorha'e theforrr exp(ia@).andtheeigenvalues
arenfi.Becausegeometrieswith azimuthalanglesequal1o@or equalto Q * 2r
areexactl)'thesamegeometries,thefu'ction exp(ia(t)shouldbeexactlythesame
asexp(ia(Q't hr 1).Thiscanonrybethecaseif a is an integer.Thus.onecon-
cludesthatonlyintegralmultiplesofficanbe'.allowed"valuesofthe:-cornponent
of angularmomenturn.Experimentalry,,onemeasuresthe:-componentof anan-
-eularmomentumbi, placingthesystempossessingthea'gularmomentumin a
magneticfieldof strengthB andobser'inghowmany---corxponentenergystates
arise.Forexample,aboronatomwith its2porbitalhasoneunitof orbitalangular
momentum.so one finds threeseparate;-co,lponent 'alues r.l,hichareusually
denotednt : -7, m :0, andnt = L Anotherexampleis offeredby thescan_
dium atomrvith oneunpairedelectronin a d orbital:this atom,sstatessplit into
five (rr = -2, -1. 0. 1,2) z--componentstates.In eachcase.onefindsZL + l
valuesof ther/rquantumnumber,and,becauser is a' integer.2L + l is anodd
integer.Both of theseobservationsareconsistentwith the expectationthatonlv
integervaluescanoccurfor Z, eigenvalues.
However.it hasbeenobservedthatsomespeciesdo notpossess3 or 5 or 7 or
9:-componentstatesbutanevennumberofsuchstates.In particular.whenelec-
trons,protons,or neutronsaresubjectedto thekind of magneticfieldexperiment
mentionedabo'e, theseparticlesare observedto haveonly two z-componenr
eigenvalues.Because,aswe discusslaterin this text.all an-qularmomentahave
.?-componenteigenvaluesthat are separatedfrom one anotherby unit muiti_
plesof fi. oneis forcedto concludethatthesethreefundamentalbuilding-block
particleshave:-componenteigenvaluesof u2ri and.-r
/2h. The appearanceof

half-integerangularmomentais notconsistentwith theobservationmadeearlier
that@andQ + 2;r correspondto exactlythe samephysicalpoint in coordinate
space,u'hich,in turn,impliesthatonlyfull-integerangularmomentaarepossible.
The resolutionof the aboveparadox(i.e.,horvcanhalf-integerangularmo-
mentaexist?)involvedrealizingthatsomeangularmomentacorrespondnotto
ther x p angularmomentaof a physicalmassrotating.but.insteadareintrinsic
propertiesof certainparticles.That is, the intrinsicangularmomentaof elec-
trons,protons,andneutronscannot be viewedasarisingfrom rotationof some
massthatcomprisestheseparticles.Insteadsuchintrinsicangularmomentaare
fundamental
"built in" characteristicsof theseparticles.For example.the trvo
I l2l1and-112h angularmomentumstatesof anelectron,usuallydenotedo and
p. respectively.aretwo internalstatesofthe electronthataredegeneratein the
absenceof a magneticfield but which representfwo distinctstatesof the elec-
tron.Analogousl,v,a protonhasIl2h and-l l2li states,asdo neutrons.All such
half-integerangularmomentumstatescan not be accountedfor usingclassical
mechanicsbut areknorvnto arisein quantummechanics.
45

Chapter1(1)

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