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