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

molecularproperties

we krow that all molcculesare made of atoms which. in turn. containnuclei and electrons.As I discussin this introcjuctory

section,the equationsthat

govern the motions of electronsand of nuclei are not the familiar Newton

equatrons.

F:ma

(l.l)

but a nervset of equationscalledSchrodingerequations.when scientistsfirst

studiedthe behaviorof electronsand nuclei.thev tried to interprettherr experimentalfindin-ssin termsof classicalNewtonianmotions.but suchatrempts

eventuallyfailed.Theyfoundthatsuchsrnalllightparticlesbehaved

in a waythat

simply is not consistent

rvith the Neu'tonequations.

Let me norvillustratesorne

ofthe experimentaldatathat gaverise to theseparadoxesand shorvyou how the

scientists

ofthoseearlytimesthenusedthesedataro suggestnew equatrons

that

theseparticlesrnight obcy.I want to stressthat the Schrcidinger

equationwas not

derivedbut postulated

by thesescientists.

In fact,to date,rlo one hasbeenable

t o d e r i r et h e S c h r c i d i n geeqr u a t i o n .

Fron.rthe pioneeringwork of Braggon ditrractionof x-raysfiom planesof

atomsor ions in crvstals,it was knownthat peaksin the intensityof ditliacted

x-rayshavin,u

wavelength

i rvouldoccurat scattering

anglesg determinedby the

larnousBragg equation:

nt : 24 tinp

(1.2)

where d is the spacingbetweenneighborin,u

planesof atomsor ions. These

quantitiesare illustratedin Fig. I . I . Therearemany suchdiffractionpeaks,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 en r( n - 1 , 2 . 3 , . . . ) . T h e B r a - s - q f o r m u l a

can be derivedby consideringwhen two photons,one scatteringfrom the second

plane in 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

Scatteringof

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 the wavelengthof the photons.

The importance

of thesex-rayscatterin_e

experiments

to the studyof electrons

and nuclei appearsin tl.reexperirnentsof Davissonand Gernter.in 1927, u'ho

scattered

electronsof(reasonably)

fixedkineticenerg)'E from metalliccrvstals.

Theseworkersfoundthatplotsofthenumberofscattered

electrons

asa functionof

"peaks"

scattering

angled displayed

atangles6 thatobeyeda Bragg-likeequation.

The startlingthing aboutthis observation

is that electronsare particles.vet the

Bragg equationis basedon the propertiesof waves.An important observation

derivedfron.rthe Davisson-Germerexperimentswas that the scatteringangles6

observedfor electronsof kinetic energ.vE couldbe fit to the Braggni, : 2d sin0

equationif a wavelengthwere ascribedto theseelectronsthat u'asdefinedby

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

(1.3)

where rl. is the massof the eiectronand i is the constantintroducedby Max

Planckand Albert Einsteinin the eariy 1900sto relatea photon'senergv,Eto

its frequencyy I'ia .E : /rt,. Theseamazingfindingswere amongthe earliestto

suggestthat electrons,u,hich had alwaysbeen viewed as particles,might have

somepropertiesusually ascribedto waves.That is. as de Broglie suggestedin

1925,an electronseemsto havea wavelengthinverselyrelatedto its momenfum,

andto displaywave-typediffraction.I shouldmentionthat analogousdiffraction

u'asalsoobservedwhenothersmalllight particles(e.g.,protons.neutrons,nuclei,

and small atomic ions) were scatteredfrom crystal planes.In all such cases,

Bragg-likediffractionis observedand the Bragg equationis found to governthe

scatteringanglesifone assignsa wavelengthto the scatteringparticleaccordingto

),: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

-

c

a

c

h

a

ot

c

c

I VisibleI

*.'{fllllrilt-Irffi

""'-*,,-onllrll

frlllill

Anarvsis

lrrn

Llf

Emission

spectrumof atomic

h y d r o g e nw i t h s o m e

lines repeatedbelow to

illustrate

t h e s e r i e st o

w h i c ht h e y b e l o n g .

Paschen

Brackett

Theobservation

thatelectrons

andothersmalllight particlesdisplayivave-like

r.vas

important

behavior

because

theseparticlesarewhatall atomsandmolecules

are made of. So, if we want to fully understandthe motions and behaviorof

molecules.rvemustbe surethat*e can adequately

describesuchpropertiesfor

theirconstitr"rents.

Because

the classical

Newtonecluations

do not containtactors

that sr-rggcst

wave propertiesfor electronsor nuclei mo'",ingfreely in space.the

abovebehaviorspresented

significantchallenges.

Anotherproblemthat arosein earlystudiesof atomsand moleculesresulted

fiom the stLrdyof the photonsemittedfrom atomsand ions that had beenheated

or otherr.iseexcited(e.g.,by electricdischarge).It was found that eachkind

of atom (i.e.,H or C or O) ernittedphotonsrvhosefrequencies

u wereof very

characteristic

values.An exampleof suchemissionspectrais shownin Fig. I .2

fbr hydrogenatoms.In the top panel,we seeall of the linesemittedwith thcir

wavelengths

indicatedin nanometers.

The other panelsshow horvtheselines

havebeenanalyzed(by scientists

rvhosenarnesareassociated)

into patternsthat

relateto the specilicenergylevelsbetweenwhich transitionsoccurto emit the

corresponding

photons.

In theearlyattemptsto rationalize

suchspectrain termsof electronicmotlous.

one describedan electronas rnovinsaboutthe atomicnuclei in circularorbits

such as shor.vn

in Fig. 1.3.A circularorbit was thoughtto be stablewhen the

outwardcentrifugalfbrcecharacterized

by radiusr. and speedu (rr.u2/r) on the

electronperf-ectlir

counterbalanced

theinwardattractit,e

Coulombforce(Ze2 l121

exertedby the nucleusof chargeZ:

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

-

.

(1 . 5 )

This equation,in turn, allows one to relatethe kinetic energy lrr.ul to the

CoulornbicenergyZe2lr, and thusto expressthe t o t a le n e r g yE o f a n o r b i t i 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

terrnsof the radiusof the orbit:

E :

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.

I

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

f

'

'

-l

Z L ' -/ t ' .

1

(1.6)

The energycharacterizingan orbit ofradius r. relatil'eto the 6 : 0 reference

of energyat r --+ 3p. becomesmore andmore negative(i.e.,lorverandlou,er)asr

becomessmaller.This relationshipbetweenoutu'ardand inu,ardforcesallou's

one to concludethat the electronshouldrnovefasteras it movescloserto the

nucleussincer'2 : Ze7l(rntr). Howeter.noq'herein this mclclel

is a conceptthat

relatesto the experimental

fact that eachatoll erltitsonll'certainkindsof photons.lt u'asbelievedthatphotoner.nission

occurreduficn an electronr.rrovirrr

irr

a largercircularorbit lostenergyandnrovedto a sn.raller

circularorbit.Hor.r'ever.

the Newtoniandynamicsthatproducedthe aboveequationr.vould

allou'orbrtsof

any radius.and henceany energy.to be follorved.Thus"it wouldappcartlratthe

electronshouldbe ableto emit photonsof any encrgyas it movedfi'omorbit to

orbit.

The breakthroughthat allowed scientistssuch as Niels Bohr to apply the

circular-orbitmodel to the observedspectraldata invol','edfirst introducinsthe

ideathat the electronhas a wavelengthand that this u'avelensthi is relatedto

its nromentumby the de Broglie equationL- hlp. The key stepin the Bohr

model u'asto also specifythat the radiusof the circularorbit be suchthat the

cilcurnference

of the ctrcle2nr equalan integer(n) multipleof theu'avelen-gth

i.

Only in this way will the electron'svu'ave

experienceconstructiveinterferenceas

the electronorbits the nucleus.Thus, the Bohr relationshipthat is analogousto

the Bragg equationthat determinesat what anglesconstructiveinterferencecan

occuris

2 t r r: n ) . .

(1.7)

Both this equationandthe analogousBraggequationare illustrationsof what we

call boundaryconditions;they are extra conditionsplacedon the wavelengthto

producesomedesiredcharacterin the resultantwave( in thesecases,constructile

interference).Of course,there remains the questionof why one must impose

theseextra conditionswhen the Neu,toniandynamicsdo not requirethem. The

resolutionof this paradoxis one of the thingsthat quantummechanicsdoes.

R e t u r n i n g t ot h ea b o v ea n a l y s i as n du s i n gl , : h l p : h l ( m v ) . 2 r r : 4 ) ' , s g

weli as the force-balanceequationm"r2 1, : Z e2I 12, one can then solvefor the

radii that stableBohr orbits obey:

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 r fyo 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

Thesetrvo resultsthen allorv one to expressthe sum ofthe k i n e t i c( l r r . r ' : ) a n c l

as

Coulombpotential(-Ze2 lrl energies

g : - ! r , , z t r ' 1 t n l1t 2 t1 ' .

) '

(1.r0)

Just as in the Bragg diflraction result,rvhich specifiedat what anglesspecial

there are many stableBohr orbits.

high intensitiesoccurredin the scattering,

value

the

integer

of

ir. Those with small n have small

each labeledby a

radii. high velocitiesand more negativetotal energies(n.b., the reference

zero of energycorrespondsto the electronat r : oc, and with v :0). So.

"allowed" that causes

only certain

it is the resultthat only certainorbits are

energiesto occur and thus only certain energiesto be observedin the emitted

photolls.

It turnedout that the Bohr formula for the energylevels(labeledby r) of

an electronmoving abouta nucleuscould be usedto explainthe discreteline

an

t o m sa n d i o n s{ i . e . .H . H e . L i - : . e t c . ) t o

e m i s s i osnp e c t roaf r l l o n e - e l e c t r o

veryhighprecision.in suchan interpretation

of theexperimental

data.oneclaims

that a photonof energy

hv=R(tlni-rlni)

(l.lr)

is emittedrvhenthe atom or ion undergoesa transitionfiom an orbit having

quantumnumberni to a lower-energy

orbit havingnf. Herethe symbolR is used

to denotethe fbllo*ing collectionof factors:

R -

I

-nt,Z-e*llltr)tl'

(1.12)

The Bohr formulafbr energylevelsdid not agreeaswell w ith the observedpattern

of emissionspectrafor speciescontainingmorethana singleelectron.However,

it doesgive a reasonable

fit, for example.to the Na atomspectraif oneexamrnes

only transitionsinvolvingthe singlevalenceelectron.The primary reasonfor

the breakdorvn

of the Bohr formulais the neglectof electron-electron

Coulomb

repulsionsin its derivation.

Nevertheless.

the success

of this modelmadeit clear

thatdiscreteemissionspectracouldonly be explainedby introducingtheconcept

thatnot all orbitswere"allowed".Only specialorbitsthatobeyeda constructiveintert-erence

conditionwerereallyaccessible

to the electron'smotions.This idea

that not all energiesrverealloweclbut only certain "quantized" energiescould

occur was essentialto achievingevena qualitativesenseof agreementr.viththe

experimentalfact that emissionspectrawerediscrete.

In summary,two experimentalobservationson the behaviorof electronsthat

were crucial to the abandonmentof Newtoniandynamicswere the observations

of electrondiffractionand of discreteemissionspectra.Both of thesefindings

seem to suggestthat electronshave some wave characteristicsand that these

waveshaveonly certainallowed(i.e.,quantized)wavelengths.

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