Download Q.M3 Home work 12 Due date 28.1.15 1 S -wave scattering

Q.M3 Home work 12
Due date 28.1.15
1 S -wave scattering
At very low energies, scattering is s-wave scattering only and the scattering amplitude
−
→
− →
f ( k , k 0 ) does not depend on the scattering angle θ.
(a) Show that
f =−
b(k)
1 + ikb(k)
with b(k) real.
Hint: Remember the Optical theorem.
(b) How is b(k) related to the s-wave scattering phase shift δ0 (k)?
(c) The scattering length b0 is the k → 0 limit of b(k). Determine b0 for the repulsive
hard sphere potential
(
(~κ)2
f or r < a
2M
V (r) =
0
f or r > a
with κ > 0
−κr
(d) What is |b0 | for the Yukawa potential V (r) = V0 e κr (κ > 0) in the Born approximation?
2 A half-infinite/half-leaky box
Consider one dimensional potential
∞
f or x < 0
u(x) =
U0 δ(x − a) f or x > 0
Where a > 0.
a) Show that the stationary states with energy E can be written:


f or x < 0
 0
sin(ka)+φ(k)
u(x) =
A sin(ka) sin(kx) f or x < 0 < a

 A sin(kx + φ(k))
f or x > a
1
(1)
q
k tan(ka)
0
where k = 2mE
] where γ0 = 2mU
, φ(k) = tan−1 [ k−γ
~2
~2
0 tan(ka)
b) Is the wave function bounded?
c) Which values of γ0 gives resonance?
d) Sketch the wave function when ka = π. Explain this solution.
3 Scattering of an Electron by a neutral atom
A slow electron of wave number k is scattered by a neutral atom of effective (maximum)
radius R, such that kR 1.
(a) Assuming that the electron-atom potential is known, explain how the relevant phase
shift δl is related to the solution of a Schroedinger equation.
(b) Give a formula for the differential scattering cross section in terms of δl and k. (If
you do not remember the formula, try to guess it using dimensional reasoning.)
(c) Explain, with a diagram of the Schroedinger-equation wave function solution, how a
non-vanishing purely attractive potential might, at a particular k, give no scattering.
(d) Explain, again with a diagram, how a potential that is attractive at short distances
but repulsive at large distances might give resonance scattering near a particular k.
(e) What is the maximum value of the total cross section at the center of the resonance?
4 spin- 21 Scattering
A beam of spin- 12 particles of mass m is scattered from a target consisting of heavy
nuclei, also of spin 21 . The interaction of a test particle with a nucleus is cS1 · S2δ 3 (X1 −
X2 ), Where c is a small constant, S1 and S2 are the test particles and nuclear spins
respectively, and X1 and X2 are their respective positions.
(a) Calculate the differential scattering cross section, averaging over the initial spin
states. What is the total cross section?
(b) If the incident test particles all have spin up along the z-axis but the nuclear
spins are oriented at random, what is the probability that after scattering the test
particles still have spin up along the z-axis?
5 Variation
Consider a spherical potential U (r) with corresponding phase shift δl .
How would the phase shift change if the potential changes as U → U +δU where δU → 0.
2