QMGreensite_merged

166 CHAPTER10. SYMMETRYANDDEGENERACY

andthesearetheonlyvalueswhichcouldresultfromanaccuratemeasurement. In
short,thevaluesofangularmomentumare”quantized”inunitsof ̄h.Essentially,itis
aconsequenceofthewavelikepropertiesofmatter,combinedwiththe 2 π-periodicity
ofangles. Denoteaneigenstateofangularmomentumby

φn(r,θ)=φn(r)einθ (10.96)

andsubstituteintothetime-independentSchrodingerequation,
[
−

̄h^2

2 m

(

∂^2

∂r^2

+

1

r

∂

∂r

−

1

h ̄^2 r^2

L^2 z

)

+V(r)

]

φn=Enφn (10.97)

Usingthefactthat
Lzφn=n ̄hφn (10.98)

theSchrodingerequationbecomes
[
−

̄h^2

2 m

(

∂^2

∂r^2

+

1

r

∂

∂r

−

n^2

r^2

)

+V(r)

]

φn(r)=Enφn(r) (10.99)

Asinthecaseoftheinfinitesquarewell,thewavefunctionmustbezerowhereV(r)=
∞,i.e.
φ(r)= 0 when r≥R (10.100)

Insidethecircle,atr≤R,theSchrodingerequationis
[
∂^2
∂r^2

+

1

r

∂

∂r

+(

2 mE

̄h^2

−

n^2

r^2

)

]

φn(r)= 0 (10.101)

Define

k^2 =

2 mE

̄h^2

ρ = kr (10.102)

Intermsoftherescaledradialcoordinateρ,thetime-independentequationbecomes
[
∂^2
∂ρ^2

+

1

ρ

∂

∂ρ

+(1−

n^2

ρ^2

)

]

φn= 0 (10.103)

Thisisoneof thestandarddifferentialequations ofmathematical physics, known
as Bessel’sEquation. Thesolutionsarethe BesselFunctions,denoted Jn(ρ),and
NeumannFunctions,Nn(ρ).TheNeumannfunctionsareinfinite,andthereforenon-
differentiablewrtxandy,atr =0. SoNeumannfunctionsdonotcorrespondto
physicalstates,andcanbediscarded. Wearethereforeleftwithsolutions

φn(r)=Jn(kr) withenergy E=

̄h^2 k^2

2 m

(10.104)