Physics 481
Spring 2006
Problem Set 11
Soeren Prell
1. Problem 9.14, page 363 in Griffith - Sequence of electric dipole transitions in hydrogen.
2. Problem 9.8, page 356 in Griffith – Relative rates of spontaneous and stimulated
emission in thermal equilibrium.
3. Consider a particle of mass m and energy E scattering from a spherical potential
(
V (x) =
−V0 if r ≤ a,
0
if r > a,
where V0 is a positive real number.
(a) Use the partial wave expansion to find the differential cross-section dσ/dΩ.
Assume that ka ¿ 1 so that only the√ l = 0 term contributes. As usual,
cos(ka) tan(ja)
k 2 = 2mE/h̄2 . [Partial answer: C0 = 4π(e−ika ) j sin(ka)−k
, where
k tan(ja)+ij
j 2 ≡ 2mE
( VE0 − 1) = k 2 ( VE0 − 1).]
h̄2
(b) Use ka ¿ 1 and assume low energy scattering (V0 /E À 1) to show that the
differential cross-section can be written as
Ã
dσ
tan ja
= a2 1 −
dΩ
ja
!2
.
(c) Now consider the limit of a very short-range potential so that ja ¿ 1. Show
that the result in part (b) reduces to the same as the Born approximation for
soft-sphere scattering (p. 414 in Griffith):
dσ
=
dΩ
Ã
2mV0 a3
3h̄2
!2
.
Due Friday, April 7, 5 pm. Scores for late problem sets will be divided by 2.
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