294 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY
Wewillpostponeapplicationofthisruleuntilwestudyscattering. However,some
examplesof thedensityof finalstatesiscalledfor. Letsbeginwiththeharmonic
oscillatorintwodimensions,wheretheenergyeigenvalueswerefoundtobe
En 1 n 2 = ̄hω(n 1 +n 2 +1)= ̄hωN (18.50)
whereN = n 1 +n 2 + 1 isan integer in therange [1,∞)and the degeneracyof
eachenergy isN-fold. Then the numberof states lyingbetweenE = ̄hωN and
E+∆E= ̄hω(N+∆N)isapproximately
n = N∆N
=
E
h ̄ω
∆E
̄hω
=
E
h ̄^2 ω^2
∆E
= g(E)∆E (18.51)
Fromthisweconcludethatforthetwo-dimensionalharmonicoscillator
g(E)=
E
̄h^2 ω^2
(18.52)
AsecondexampleisthatofaparticleinacubicalboxoflengthL. Inthiscase,
proceedingasinourdiscussionofthefreeelectrongas,
En 1 n 2 n 3 =
̄h^2
2 m
π^2
L^2
(n^21 +n^22 +n^23 )
=
π^2 h ̄^2
2 mL^2
(n^21 +n^22 +n^23 )
=
π^2 h ̄^2
2 mL^2
R^2 (18.53)
ThenumberofstatesnwhichliebetweenRandR+∆Risgivenbythevolumeof
anoctantofasphericalshell
n =
1
8
[
4
3
π(R+∆R)^3 −
4
3
πR^3
]
=
1
2
πR^2 ∆R+O(∆R^3 ) (18.54)
Butfrom(18.53)
R=
(
2 mL^2
π^2 ̄h^2
) 1 / 2
E^1 /^2 (18.55)
soalso
∆R=
1
2
(
2 mL^2
π^2 ̄h^2
) 1 / 2
E−^1 /^2 ∆E (18.56)