763313A QUANTUM MECHANICS II Excercise 12
spring 2017
1. Consider the case of low-energy scattering from a spherical delta-function
shell:
V (r) = αδ(r − a),
where α and a are constants. Calculate the scattering amplitude f (θ),
the differential cross-section D(θ), and the total cross-section σ. Assume ka 1, so that only the l = 0 term contributes significantly. (To simplify matters, throw out all l 6= 0 terms right from the
start.) The main problem, of course, is to determine a0 . Express your
answer in terms of the dimensionless quantity β ≡ 2maα/h̄2 . Answer :
σ = 4πa2 β 2 /(1 + β 2 ).
2. Prove the optical theorem, which relates the total cross-section to
the imaginary part of the forward scattering amplitude:
σ=
4π
Im(f (0)).
k
3. A particle of mass m and energy E is incident from the left on the
potential


x < −a,
 0,
V (x) =  −V0 , −a ≤ x ≤ 0,

∞,
x > 0.
√
a) If the incoming wave is Aeikx (where k = 2mE/h̄), find the reflected wave. Answer :
Ae−2ika
q
k − ik 0 cot(k 0 a) −ikx
0
e
,
where
k
=
2m(E + V0 )/h̄.
k + ik 0 cot(k 0 a)
b) Confirm that the reflected wave has the same amplitude as the
incident wave.
c) Find the phase shift δ for a very deep well (E V0 ). Answer :
δ = −ka.
4. What are the partial wave phase shifts (δl ) for hard-sphere scattering?
5. For the potential in Problem 1.,
a) calculate f (θ), D(θ), and σ, in the low-energy Born approximation;
b) calculate f (θ) for arbitrary energies, in the Born approximation;
c) show that your results are consistent with the answer to Problem 1.,
in appropriate regime.