QMGreensite_merged

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)