Section5:TheHydrogenAtom

Intheseslideswewillcover:

• The3DSchrödingerequation

• Particleinaboxanddegeneratequantumstates

• Radialandangularsolutionsforcentralpotentials

• Identificationoftheangularpiecewiththeangularmomentumeigenfunctions (sphericalharmonics)

• Infinitesphericalwell

• SolutionoftheradialequationfortheCoulombpotential

• Eigenfunctions ofthehydrogenatom

• Quantumnumbersofthehydrogenatom

Particleinabox

The3DSchrödingerEquation

• Wenowmovefromparticlesin1Dspace,to3Dspace

• Thederivativein1dimension, !"

!#",isreplacedbyits3D

equivalent,the“Laplacian”operator𝛻% = '"

'#"+ '"

')"+ '"

'*"

• The3Dtime-independentSchrödingerequationfortheenergyeigenfunctions 𝜓(𝑥, 𝑦, 𝑧) andeigenvalues𝐸 ishence:

• Thenormalisation conditionforthe3Dwavefunction is∫ 𝑑𝑥565 ∫ 𝑑𝑦5

65 ∫ 𝑑𝑧565 𝜓(𝑥, 𝑦, 𝑧) % = 1

−ℏ%

2𝑚𝜕%𝜓𝜕𝑥% +

𝜕%𝜓𝜕𝑦% +

𝜕%𝜓𝜕𝑧% + 𝑉 𝑥, 𝑦, 𝑧 𝜓 𝑥, 𝑦, 𝑧 = 𝐸𝜓 𝑥, 𝑦, 𝑧

Particleinabox

Particleinabox

• Agoodexampleisa“particleinabox”withpotential

• Welookforaseparablesolution

• Wefind,bysubstitutingthistrialsolutioninthe3DSchrödingerequationandre-arranging,

𝑉 𝑥 = ?0, 𝑥 < 𝐿, 𝑦 < 𝐿, 𝑧 < 𝐿∞, outsidethebox

𝜓 𝑥, 𝑦, 𝑧 = 𝑓 𝑥 𝑔 𝑦 ℎ(𝑧)

−ℏ%

2𝑚1

𝑓 𝑥𝑑%𝑓𝑑𝑥% +

1𝑔 𝑦

𝑑%𝑔𝑑𝑦% +

1ℎ 𝑧

𝑑%ℎ𝑑𝑧% = 𝐸

Particleinabox

Particleinabox

• Thatlastequationagain:

• Wecansolvethisequationusinganicepieceoflogic:§ eachterminthesquaredbracketonlydependson1variable(𝑥,𝑦 or𝑧)

§ eachofthosevariablesisfreetovaryindependentlyfromtheothers

§ inthiscase,theonlywaytomakethetermsalwayssumtoaconstantisifeachindividualterminthesquaredbracketisseparatelyaconstant

• Wefindthateachseparatefunction𝑓, 𝑔, ℎ satisfiesthe1DSchrödingerequationfortheinfinitepotentialwell.Wemultiplythemtogetherforthetotalsolution,𝜓 = 𝑓 Q 𝑔 Q ℎ

−ℏ%

2𝑚1

𝑓 𝑥𝑑%𝑓𝑑𝑥% +

1𝑔 𝑦

𝑑%𝑔𝑑𝑦% +

1ℎ 𝑧

𝑑%ℎ𝑑𝑧% = 𝐸

Particleinabox

Particleinabox

• Summarising,theparticleinaboxhasenergyeigenfunctionslabelledbythreequantumnumbers(𝑛#, 𝑛), 𝑛*)

• 𝜓STU(𝑥) arethedifferentstates𝑛 whicharesolutionsofthe1Dinfinitepotentialwell

• Theenergyeigenvaluesarethenthesumofthecorrespondingeigenvaluesforthe1Dinfinitepotentialwell,

• Wenoticeherethatdistincteigenfunctions canhavethesameenergy(degeneratestates),e.g.(2,1,1),(1,2,1) and(1,1,2)

𝜓VU 𝑥, 𝑦, 𝑧 = 𝜓SWTU 𝑥 𝜓SX

TU 𝑦 𝜓SYTU(𝑧)

𝐸(𝑛#, 𝑛), 𝑛*) =𝜋%ℏ%

8𝑚𝐿% 𝑛#% + 𝑛)% + 𝑛*%

Centralpotentials

Radialandangularsolutionsforcentralpotentials

• Acentralpotentialdependsonlyonthedistancefromtheorigin,𝑉 𝑥, 𝑦, 𝑧 = 𝑉(𝑟).Animportantexampleis,thepotentialaroundahydrogennucleus(proton)attheorigin

𝑟

+𝑒

−𝑒(Sorryfortheillustration

withclassicalpointparticles.Theyareofcourse

probabilitycloudsinQM!)

𝑥 𝑦

𝑧

Centralpotentials

Radialandangularsolutionsforcentralpotentials

• Acentralpotentialdependsonlyonthedistancefromtheorigin,𝑉 𝑥, 𝑦, 𝑧 = 𝑉(𝑟).Animportantexampleis,thepotentialaroundahydrogennucleus(proton)attheorigin

• Theseproblemsarebesttreatedusingsphericalpolarco-ordinates,(𝑟, 𝜃, 𝜙)

• Non-examinable:weneedtheexpressionfortheLaplacianinsphericalpolarco-ordinates(ouch!):

𝛻% =1𝑟%

𝜕𝜕𝑟 𝑟%

𝜕𝜕𝑟 +

1𝑟% sin 𝜃

𝜕𝜕𝜃 sin 𝜃

𝜕𝜕𝜃 +

1𝑟%sin%𝜃

𝜕%

𝜕𝜙%

Centralpotentials

Radialandangularsolutionsforcentralpotentials

• Non-examinable:Weagainlookforaseparablesolutionfortheenergyeigenfunctions,

• Substitutingthisinthe3DSchrödingerequationandre-arranging,wefind:

• Byasimilarlogictothecubicalboxsolution… theleft-handsideofthisequationisafunctionof(𝜃, 𝜙) onlyandtheright-handsideisafunctionof𝑟 only.Buttheymustremainequalas(𝑟, 𝜃, 𝜙) varyindependently.Sotheymustbothbeequaltothesameconstant,let’scallit𝜆

𝜓 𝑟, 𝜃, 𝜙 = 𝑅 𝑟 𝑌(𝜃, 𝜙)

−ℏ%

𝑌1

sin 𝜃𝜕𝜕𝜃 sin 𝜃

𝜕𝑌𝜕𝜃 +

1sin%𝜃

𝜕%𝑌𝜕𝜙% =

ℏ%

𝑅𝜕𝜕𝑟 𝑟%

𝜕𝑅𝜕𝑟 + 2𝜇 𝐸 − 𝑉 𝑟 𝑟%

Switchingtousing𝜇 formass,since𝑚 isabouttomeansomethingelse

Centralpotentials

Radialandangularsolutionsforcentralpotentials

• Theleft-handsideofthisequationbecomes,

• Aha!Theleft-handsideisjusttheoperatorforthetotalangularmomentum,𝐿e%,fromthepreviousSection:

• Hence,theangularpieceoftheenergyeigenfunctions foracentralpotentialisjustthesphericalharmonicfunctions𝑌fg(𝜃, 𝜙) where𝜆 = 𝑙 𝑙 + 1 ℏ% – that’sconvenient

−ℏ%1

sin 𝜃𝜕𝜕𝜃 sin 𝜃

𝜕𝜕𝜃 +

1sin%𝜃

𝜕%

𝜕𝜙% 𝑌 𝜃, 𝜙 = 𝜆𝑌 𝜃, 𝜙

𝐿e%𝑌 = 𝑙 𝑙 + 1 ℏ%𝑌

Centralpotentials

Radialandangularsolutionsforcentralpotentials

• Substituting𝜆 = 𝑙 𝑙 + 1 ℏ% intheradialpiece,wefind

• Ausefulchangeofvariablesistosubstitute𝑢 𝑟 = 𝑟 Q 𝑅(𝑟)[i.e.𝑅 = 𝑢/𝑟].Inthiscase,

• Wenotethatthislooksjustlikea1DSchrödingerequation,excepttheusualpotentialhasbeenreplacedbyan“effective

potential” 𝑉 𝑟 + f fkT ℏ"

%lm"

−ℏ%

2𝜇1𝑟%

𝑑𝑑𝑟 𝑟%

𝑑𝑅𝑑𝑟 + 𝑉 𝑟 +

𝑙 𝑙 + 1 ℏ%

2𝜇𝑟% 𝑅 = 𝐸𝑅

−ℏ%

2𝜇𝑑%𝑢𝑑𝑟% + 𝑉 𝑟 +

𝑙 𝑙 + 1 ℏ%

2𝜇𝑟% 𝑢 = 𝐸𝑢

[Again,𝜇 =mass]

Note:thisequationdependsontheangularmomentumstate,through𝑙

Centralpotentials

Infinitesphericalwell

• Agoodexampleofthisisa“particleinasphere”,where

• Inthiscase,theenergyeigenfunctions 𝑢S(𝑟) at𝑟 < 𝑎 satisfy

• Boundaryconditionis𝑢 = 0 at𝑟 = 𝑎.Thefirsteigenfunctionsare𝑢 ∝ sin 𝑘𝑟 for𝑙 = 0 and𝑢 ∝ qrs tm

tm− cos 𝑘𝑟 for𝑙 = 1

• Notethatthefullenergyeigenfunction alsoincludestheangularpiece,𝜓 𝑟, 𝜃, 𝜙 = [𝑢S 𝑟 /𝑟]𝑌fg(𝜃, 𝜙) – thestateoftheparticleischaracterisedby3quantumnumbers𝑛, 𝑙,𝑚

𝑉 𝑟 = ?0, 𝑟 < 𝑎∞, 𝑟 > 𝑎

−ℏ%

2𝜇𝑑%𝑢S𝑑𝑟% +

𝑙 𝑙 + 1 ℏ%

2𝜇𝑟% 𝑢S = 𝐸𝑢S

Centralpotentials

Infinitesphericalwell

• Herearethefirstfewradialeigenfunctions fortwovaluesof𝑙:

• Theprobabilityoffindingtheparticlebetween𝑟 and𝑟 + 𝑑𝑟 isgivenby𝑟% 𝜓 % ∝ 𝑟% 𝑅 % ∝ 𝑢 % – theextrafactorof𝑟% isappearingbecauseofthevolumeelementinsphericalpolars

𝑙 = 0 𝑙 = 1𝑢(𝑟) (unnormalised) 𝑢(𝑟) (unnormalised)

𝑟/𝑎 𝑟/𝑎

edgeedge

Eigenfunctions ofthehydrogenatom

SolutionofradialequationforCoulombpotential

• However,themostimportantexampleofacentralpotentialisanelectroninahydrogenatomorbitingaroundanucleus,whichfollowstheCoulomb

potentialenergy,𝑉 𝑟 = − y"

z{|}m

• Inthiscasetheradialequationbecomes(where𝑎 = 4𝜋𝜀�ℏ%/𝜇𝑒%),

−𝑑%𝑢S𝑑𝑟% + −

2𝑎𝑟 +

𝑙 𝑙 + 1𝑟% 𝑢S =

2𝜇𝐸ℏ% 𝑢S

𝑟

+𝑒

−𝑒

(Sorryfortheillustrationwithclassicalpointparticles.

TheyareofcourseprobabilitycloudsinQM!)

Eigenfunctions ofthehydrogenatom

SolutionofradialequationforCoulombpotential

• Wecansolvethisequationbyconsideringtwolimits…

• If𝑟 → ∞,wehave−!"�!m"

= %l�ℏ"𝑢.Werecognisethisequation

ashavingsolution𝑢(𝑟) ∝ 𝑒6m/�,where𝐸 = − ℏ"

%l�"

• If𝑟 → 0,wehave−!"�!m"

+ f(fkT)m"

𝑢 = 0.Thisequationhasasolution𝑢(𝑟) ∝ 𝑟fkT

• Thislogicmotivatesthatthecompletesolutionisapolynomialmultipliedbyanexponential,dependingonthevalueof𝑙

Eigenfunctions ofthehydrogenatom

• Thetextbookexplains[wewon’tgivethederivationheresinceit’sabitinvolved…] thattheenergyeigenvalues𝐸S canbecharacterisedbyasinglequantumnumber𝑛,andagivenenergyhas𝑛degenerateeigenfunctions 𝑢Sf(𝑟) givenby𝑙 = 0,1, … , 𝑛 − 1

• Herearesomesolutionsfor𝑅Sf 𝑟 = 𝑢Sf(𝑟)/𝑟 correspondingtothefirsttwovaluesof𝑛 = 1,2 andtheallowedvaluesof𝑙:

𝑅T� =2𝑎V/%

𝑒6m/�

𝑛 = 1 𝑛 = 2

𝑙 = 0

𝑙 = 1

𝑅%� =1

2� 𝑎V/%1 −

𝑟2𝑎 𝑒6m/%�

𝑅%T =1

2 6� 𝑎V/%𝑟𝑎 𝑒6m/%�(notallowed)

Eigenfunctions forhydrogenatom

Eigenfunctions ofthehydrogenatom

• Theradialextentoftheeigenfunctions arecharacterisedbythe

parameter𝑎 = z{|}ℏ"

ly"=

5.3×106TT𝑚 knownastheBohrradius

• Herearethefirstfewradialeigenfunctions,plottedasaprobabilitydensity 𝑢Sf % =𝑟% 𝑅Sf %

Eigenfunctions forhydrogenatom

Eigenfunctions ofthehydrogenatom

Energylevelsandquantumnumbersofhydrogenatom

• Thetextbookshowsthattheenergyeigenvaluesonlydepend

on𝑛,andaregivenby𝐸S = −𝐸T/𝑛%,where𝐸T =l%ℏ"

y"

z{|}

%.

ThisisthefamousBohrformulaforthehydrogenenergylevels

Imagecredit:Hyperphysics

Eigenfunctions ofthehydrogenatom

Energylevelsandquantumnumbersofhydrogenatom

• Let’srecapthefullargument:

1. TheSchrödingerequationforacentralpotentialalwayshasseparablesolutions𝜓(𝑟, 𝜃, 𝜙) = 𝑅Sf 𝑟 𝑌fg(𝜃, 𝜙),where𝑌fg arethesphericalharmonicsand𝑅Sf satisfiesaradialequation

2. Hence,theseareautomaticallyeigenfunctions ofangularmomentum,with𝐿% = 𝑙(𝑙 + 1)ℏ% and𝐿* = 𝑚ℏ (−𝑙 ≤ 𝑚 ≤ +𝑙)

3. ForaCoulombpotentialinparticular,theenergyeigenvaluesdonotdependon𝑙,butonlyonasinglequantumnumber𝑛,𝐸S =− 𝐸T/𝑛%,withtherestriction0 ≤ 𝑙 ≤ 𝑛 − 1

4. Therearehence3quantumnumberswhichcharacterise thehydrogenatom:totalenergy(𝑛),totalangularmomentum(𝑙)andthe𝑧-componentofangularmomentum(𝑚)

Eigenfunctions ofthehydrogenatom

Energylevelsandquantumnumbersofhydrogenatom

• Thefulleigenfunctions 𝜓(𝑟, 𝜃, 𝜙) areacombinationoftheradialeigenfunction andsphericalharmonics:

Eigenfunctions ofthehydrogenatom

Energylevelsandquantumnumbersofhydrogenatom

• Countingthenumberofdistincteigenstatesassociatedwithaspecificenergy𝑛 …

• Thereare𝑛 possiblevaluesof𝑙

• Foreach𝑙,thereare(2𝑙 + 1) allowedvaluesof𝑚

• Combiningallthose,wefind𝑛% distinctstates

Eigenfunctions ofthehydrogenatom

Summary

• Thehydrogenatomhasdiscreteenergylevels𝐸S = −𝐸T/𝑛%,

where𝐸T =l%ℏ"

y"

z{|}

%and𝒏 istheenergyquantumnumber

• Foreachstate𝑛,thereare𝑛 differentvaluesof𝑙 =(0,1, … , 𝑛 − 1),where𝒍 isthequantumnumberfortotalangularmomentum and𝐿% = 𝑙(𝑙 + 1)ℏ%

• Foreachstate𝑙,thereare(2𝑙 + 1) differentvaluesof𝑚 =(−𝑙, −𝑙 + 1,… , 𝑙 − 1, 𝑙),where𝒎 isthequantumnumberforthe𝒛-componentofangularmomentum and𝐿* = 𝑚ℏ

• Theenergyeigenfunctions arethen𝜓Sfg 𝑟, 𝜃, 𝜙 =𝑅Sf 𝑟 𝑌fg 𝜃, 𝜙 = ���(m)

m𝑌fg 𝜃, 𝜙

Summary

Eigenfunctions ofthehydrogen

atom

Centralpotentials

• ForaCoulombpotential,theradialpiecetakestheformofapolynomialmultipliedbyanexponential

• Thehydrogenatomischaracterisedbyquantumnumbers(𝑛, 𝑙, 𝑚) for(𝐸, 𝐿%, 𝐿*) where0 ≤ 𝑙 ≤ 𝑛 − 1and−𝑙 ≤ 𝑚 ≤ 𝑙

• Ifthepotentialdependsonlyontheradius𝑟,thentheangularpieceof𝜓 istheangularmomentumeigenfunctions,thesphericalharmonics𝑌fg(𝜃, 𝜙)

• Theradialpieceof𝜓 satisfiesthe1DSchrödingerequationforaneffectivepotential,dependingon𝑙

• The3Dtime-independentSchrödingerequationis

− ℏ"

%g𝛻%𝜓 + 𝑉𝜓 = 𝐸𝜓,where𝛻% = '"

'#"+ '"

')"+ '"

'*"

• Aconvenientmethodofsolvingistosearchforawavefunction separableinthesystemco-ordinates

• Differenteigenfunctions whichhavethesameenergyarecalleddegeneratestates

Particleinabox