Chapter 1 The basics of quantum mechanics

C h a p t e r1

The basicsof quantum mechanics

1.1 Why quantum mechanicsis necessaryfor describing m o l e c u l a rp r o p e r t i e s

we krow that all molcculesare madeof atomswhich. in turn. containnuclei and electronsA. s I discussin this introcjuctorsyection,the equationsthat governthe motionsof electronsand of nuclei are not the familiar Newton equatrons.

F:ma

(l.l)

but a nervsetof equationscalledSchrodingeer quationsw. hen scientistsfirst studiedthe behaviorof electronsand nuclei.thev tried to interprettherrexperimentaflindin-ssin termsof classicaNl ewtonianmotions.but suchatrempts eventuallyfailed.Theyfoundthatsuchsrnall ightparticlesbehavedin a waythat simplyis not consistenrtviththe Neu'tonequationsL.et me norvillustratesorne ofthe experimentadl atathatgaveriseto theseparadoxesandshorvyou how the scientistosfthoseearlytimesthenusedthesedataro suggesnt ewequatrontshat theseparticlesrnightobcy.I wantto stressthatthe Schrcidingeerquationwasnot derivedbut postulatedby thesescientistsI.n fact,to date,rlo onehasbeenable t o d e r i r et h eS c h r c i d i n geeqr u a t i o n .

Fron.trhe pioneeringwork of Braggon ditrractionof x-raysfiom planesof atomsor ionsin crvstalsi,t wasknownthatpeaksin the intensityof ditliacted x-rayshavin,uwavelengthi rvouldoccurat scatterinagnglesg determinedby the larnousBraggequation:

nt : 24 tinp

(1.2)

whered is the spacingbetweenneighborin,pulanesof atomsor ions.These quantitiesareillustratedin Fig. I .I . Therearemanysuchdiffractionpeaks,each l a b e l e d b y a d i f f e r e n t v a l utehoefi n r e g enr( n - 1 , 2 . 3 , . . . ) . T h e B r a - s - q f o r m u l a

canbe derivedby consideringwhentwo photons,onescatteringfrom the second planein the figure and the secondscatteringfrom the third plane,will undergo constructiveinterference.This condition is met when the ,.extrapath leneth,'

T h e b a s i c so f o u a n t u mm e c h a n i c s

S c a t t e r i n go f t w o b e a m sa t a n g l e6 f r o m t w o p l a n e si n a crystalspacedby d.

coveredby the secondphoton(i.e.,the lengthfront pointsA to B to C; is an integermultipleof thewavelengthof thephotons.

Theimportanceof thesex-rayscatterine_xeperimenttso thestudyof electrons and nuclei appearsin tl.reexperirnentsof Davissonand Gernter.in 1927,u'ho scattereedlectronsof(reasonablyf)ixedkineticenerg)E' from metalliccrvstals. Theseworkersfoundthatplotsofthenumberofscattereedlectronassa functionof scatterinagngled displayed"peaks"atangles6 thatobeyeda Bragg-ilkeequation. The startlingthingaboutthis observationis thatelectronsareparticlesv. et the Bragg equationis 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

)":hl(2m"E)t'2.

(1.3)

whererl. is the massof the eiectronand i is the constantintroducedby Max PlanckandAlbert Einsteinin the eariy 1900sto relatea photon'senergv,Eto its frequencyy I'ia .E : /rt,. Theseamazingfindingswereamongthe earliestto suggest hat electrons,u,hich had alwaysbeenviewedas particles,might have somepropertiesusuallyascribedto waves.That is. as de Broglie suggestedin 1925,an electronseemsto havea wavelengthinverselyrelatedto its momenfum, andto displaywave-typediffraction.I shouldmentionthatanalogousdiffraction u'asalsoobservedwhenothersmalllight particles(e.g.,protons.neutronsn, uclei, and small atomic ions) were scatteredfrom crystal planes.In all such cases, Bragg-likediffractionis observedandthe Braggequationis foundto governthe scatteringanglesifone assignsawavelengthto thescatteringparticleaccordingto

),:hlQmE)1t2,

(1.4)

where rl is the mass of the scatteredparticle and /r is Planck's consranr ( 6 . 6 2x l 0 - 2 7e r gs ) .

W h y q u a n t u mm e c h a n i c si s n e c e s s a rfyo r d e s c r i b i n gm o l e c u l a rp r o p e r t i e s

l,.inm

a

c

o

c

-

c

h

a

t

c

*.'{flIVlilsilblerI ilt-Irffi

frlllill ""'-*,,-onllrll

Anarlvrsrnis Paschen Llf Brackett

Emission spectrumof atomic h y d r o g e nw i t h s o m e linesrepeatedbelow to i l l u s t r a t et h e s e r i e st o whichthey belong.

T h eo b s e r v a t i otnh a te l e c t r o nas n do t h e rs m a l ll i g h tp a r t i c l e sd i s p l a yi v a v e - l i k e behavior.vaismportanbt ecausetheseparticlesarewhatall atomsandmolecules are made of. So, if we want to fully understandthe motions and behaviorof moleculesr.vemustbe surethat*e canadequateldyescribesuchpropertiesfor theirconstitr"renBtse.causetheclassicaNl ewtonecluationdso notcontaintactors that sr-rggcswtavepropertiesfor electronsor nuclei mo'",ingfreely in space.the abovebehaviorspresentesdignificanct hallenges.

Anotherproblemthatarosein earlystudiesof atomsandmoleculesresulted fiom the stLrdyof the photonsemittedfrom atomsand ionsthat hadbeenheated or otherr.iseexcited(e.g.,by electricdischarge)I.t was found that eachkind of atom(i.e.,H or C or O) ernittedphotonsrvhosefrequencieus wereof very characteristivcaluesA. n exampleof suchemissionspectrais shownin Fig. I .2 fbr hydrogenatoms.In the top panel,we seeall of the linesemittedwith thcir wavelengthisndicatedin nanometersT.he otherpanelsshowhorvtheselines havebeenanalyzed(by scientistrsvhosenarnesareassociatedin) topatternsthat relateto the specilicenergylevelsbetweenwhich transitionsoccurto emit the c o r r e s p o n d i npgh o t o n s .

In theearlyattemptsto rationalizseuchspectrain termsof electronicmotlous. one describedan electronas rnovinsaboutthe atomicnucleiin circularorbits suchas shor.vnin Fig. 1.3.A circularorbit was thoughtto be stablewhenthe outwardcentrifugaflbrcecharacterizebdy radiusr. andspeedu (rr.u2/r) on the electronperf-ectlcior unterbalancethdeinwardattractit,Ce oulombforce(Ze2 l121 exertedby thenucleusof chargeZ:

n.r- ,'t'= Ze-lt -.

(1.5)

This equation,in turn, allows one to relatethe kinetic energylrr.ul to the CoulornbicenergyZe2lr, andthusto expressthe t o t a le n e r g yE o f a n o r b i ti n

T h e b a s i c so f q u a n t u mm e c h a n i c s

C h a r a c t e r i z a t i o no f small and large stable orbits of radii 11 and 12 for an electron moving arouno a nucleus.

terrnsof theradiusof theorbit:

E :

-I r t t . . r -- Z t ' - , r ' :

-l - Z L ' -/ t ' .

f

'

'

1

(1.6)

The energycharacterizingan orbit ofradius r. relatil'eto the 6 : 0 reference of energyatr --+ 3p.becomesmoreandmorenegative(i.e.,lorverandlou,er)asr becomessmaller.This relationshipbetweenoutu'ardand inu,ardforcesallou's oneto concludethat the electronshouldrnovefasteras it movescloserto the nucleussincer'2: Ze7l(rntr). Howeter.noq'herein thismclcleils a concepthat relatesto the experimentaflactthateachatoll erltitsonll'certainkindsof photons.lt u'asbelievedthatphotoner.nissioonccurreduficn an electronr.rrovirrirr a largercircularorbitlostenergyandnrovedto a sn.rallecrircularorbit.Hor.r'ever. t h e N e w t o n i a nd y n a m i c st h a tp r o d u c e dt h e a b o v ee q u a t i o nr . v o u l da l l o u 'o r b r t so f anyradius.andhenceanyenergyt.o be follorvedT. hus"it wouldappcartlratthe electronshouldbe ableto emitphotonsof anyencrgyasit movedfi'omorbit to orbit.

The breakthroughthat allowedscientistssuchas Niels Bohr to apply the circular-orbitmodel to the observedspectraldatainvol','edfirst introducinsthe ideathatthe electronhasa wavelengthandthatthis u'avelensthi is relatedto its nromentumby the de BroglieequationL- hlp. The key stepin the Bohr modelu'asto alsospecifythatthe radiusof the circularorbit be suchthatthe cilcurnferencoef the ctrcle2nr equalanintege(rn) multipleof theu'avelen-git.h Only in this way will the electron'svu'avexperienceconstructiveinterferenceas the electronorbitsthe nucleus.Thus,the Bohr relationshipthat is analogousto the Braggequationthatdeterminesat what anglesconstructiveinterferencecan o c c u ri s

2trr: n)..

(1.7)

Both this equationandthe analogousBraggequationareillustrationsof whatwe call boundaryconditions;they are extraconditionsplacedon the wavelengthto producesomedesiredcharacterin theresultantwave(in thesecasesc, onstructile interference)O. f course,there remainsthe questionof why one must impose theseextraconditionswhen the Neu,toniandynamicsdo not requirethem.The resolutionof this paradoxis oneof the thingsthatquantummechanicsdoes.

R e t u r n i n g t ot h ea b o v ea n a l y s iasn du s i n gl , : h l p : h l ( m v ) . 2 r r : 4 ) ' , s g weli asthe force-balance quationm"r2 1, : Z e2I 12, onecanthensolvefor the radii that stableBohr orbitsobey:

y :1nhl2n)2 /(m"Ze2)

(l 8)

and, in turn. for the velocities ofelectrons in these orbits,

v : ze2lfuhl2tr).

(l.e)

W h y q u a n t u mm e c h a n i c si s n e c e s s a rfyo r d e s c r i b i n gm o l e c u l a rp r o p e r t i e s

Thesetrvoresultsthen allorvoneto expressthe sum ofthe k i n e t i c( l r r . r ' : ) a n c l Coulombpotentia(l-Ze2 lrl energieas s

g :

-!r,,zt )'

r ' 1 t n l1t 2 t1 ' .

(1.r0)

Justas in the Bragg diflraction result,rvhich specifiedat what anglesspecial high intensitiesoccurredin the scatteringt,hereare many stableBohr orbits. each labeledby a value of the integerir. Those with small n have small 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 occur and thus only certainenergiesto be observedin the emitted photolls.

It turnedout that the Bohr formula for the energylevels(labeledby r) of an electronmovingabouta nucleuscould be usedto explainthe discreteline e m i s s i osnp e c t roaf r l l o n e - e l e c t r oa nt o m sa n di o n s{ i . e . .H . H e . L i - : . e t c . ) t o veryhighprecisioni.n suchaninterpretatioonf theexperimentadlata.oneclaims thata photonof energy

hv=R(tlni-rlni)

(l.lr)

is emittedrvhenthe atom or ion undergoesa transitionfiom an orbit having quantumnumberni to a lower-energoyrbithavingnf. HerethesymbolR is used to denotethe fbllo*ing collectionof factors:

R - -I n t , Z - e * l l l t r ) t l '

(1.12)

TheBohr formulafbr energylevelsdid notagreeaswell w ith theobservedpattern of emissionspectrafor speciescontainingmorethana singleelectronH. owever, it doesgivea reasonablfeit, for examplet.o theNa atomspectraif oneexamrnes only transitionsinvolvingthe singlevalenceelectron.The primaryreasonfor thebreakdorvonf the Bohrformulais the neglecot f electron-electroCnoulomb repulsionsin itsderivationN. everthelestsh.esuccesosf thismodelmadeit clear thatdiscreteemissionspectracouldonlybeexplainedby introducingtheconcept thatnot all orbitswere"allowed".Only speciaol rbitsthatobeyeda constructiveintert-erenceonditionwerereallyaccessiblteo theelectron'smotions.Thisidea that not all energiesrverealloweclbut only certain"quantized"energiescould occurwas essentiatlo achievingevena qualitativesenseof agreemenrt.viththe experimentaflact that emissionspectrawerediscrete.

In summary,two experimentaol bservationson the behaviorof electronsthat werecrucial to the abandonmenot f Newtoniandynamicswerethe observations of electrondiffractionand of discreteemissionspectraB. oth of thesefindings seemto suggestthat electronshavesome wave characteristicsand that these waveshaveonly certainallowed(i.e.,quantized)wavelengths.

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