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).