QMGreensite_merged

Chapter 12

The Hydrogen Atom

Historically, thefirstapplicationoftheSchrodingerequation(bySchrodingerhim-
self)wastothe Hydrogenatom,andtheresultforthefrequencyoftheHydrogen
spectrallineswasthe sameasthat obtainedbyBohr.^1 Inthislecture wewillre-
traceSchrodinger’sstepsinsolvingtheSchrodingerequationfortheHydrogenatom.
Unlikethecaseoftheharmonicoscillator,andcaseofangularmomentum,theraising-
loweringoperatortrickdoesn’tapplyhere.Nevertheless,wewillseethattheenergy
eigenvaluesoftheHamiltonianareagaindetermined,ultimately,byalgebra.
TheHydrogenatomisasimplesystem:alightelectronboundtoaheavyproton.
Theprotonissomuchheavierthantheelectron(about 2000 timesheavier)thatwe
mayregarditspositionasfixed,atthecenterofasystemofsphericalcoordinates.
Thepotentialenergybetweentheelectronandprotonisgiven,insuitableunits,by
theCoulombexpression

V(r)=−

e^2

r

(12.1)

Sincethispotentialissphericallysymmetric,weknowfromtheprecedinglecturethat
theHamiltonianwillcommutewiththeangularmomentumoperators,andtherefore
thatenergyeigenstatesmayalsobechosentobeeigenstatesofL^2 andLz,i.e.

φE(r,θ,φ)=Rkl(r)Ylm(θ,φ) (12.2)

Thetime-independentSchrodingerequationis
[
−

̄h^2

2 m

∇^2 −

e^2

r

]

φE(r,θ,φ)=EφE(r,θ,φ) (12.3)

or [
1
2 m

(p ̃^2 r+

1

r^2

L ̃^2 )−e

2

r

]

Rkl(r)Ylm(θ,φ)=ERklYlm (12.4)

(^1) ThedifferenceisthatBohr’sratheradhocquantizationcondition(mvr=n ̄h)wassuccessful
onlyforHydrogenandafewothersimpleatoms.Incontrast,theSchrodingerequationisalawof
motion,asfundamentalasF=ma(whichitreplaces).

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