Physics 212 – Problem Set 11 – Spring 2010
1. Scattering in 1-D.
(a) Consider a single potential step. The potential is zero from x < 0 and a constant V0 for x > 0. A particle
of mass m is incoming from x < 0 with energy E = h̄ω.
i. Write the Schrodinger equation for x < 0. Write the Schrodinger equation for x > 0. Write expressions
for the general solution in the region where x < 0 and in the region where x > 0.
ii. Remembering that the particle is incoming from x < 0, find the probability current for the x < 0
region. Find the probability current for the x > 0 region. From this find the probability of transmission
jref
(= jtrans
jinc ) and the probability of reflection (= jinc ) in terms of the relevant wave numbers and unknown
coefficients.
(b) The general solution for an arbitrary V (x) may be written Ψ(x, t) = |Ψ(x, t)|eiS(x,t)/h̄ . Find the probability
current associated with this solution. Notice how the current depends upon the phase S(x, t)
(c) Suppose we could write the general solution for a potential V (x) as Ψ(x, t) = AeiS(x,t)/h̄ , where A is a
constant. Plug this into the time-dependent Schrodinger equation to find an equation for S(x, t). Now let
S(x, t) = W (x) − Et. Find the equation for W (x).
(d) Now we will do a semiclassical (or WKB) approximation: Expand W (x) as a series in h̄:
W = W (0) + h̄W (1) + · · · =
∞
X
h̄i W (i) .
i=0
Under what conditions might this be a reasonable thing to do? Plug this in and match terms of the same
order in h̄ to find equations for W (0) and W (1) .
(e) Solve the two equations above to show that to first order in h̄ we have solutions
Z x
−1/2
0
0
ψ(x) ∼ k
exp ±i
k(x )dx
,
x0
where k(x) =
p
2m(E − V (x)) and x0 is some initial point.
2. A few intermediate steps
ikr
(a) Verify our result for the scattered wave current obtained from φsc = f (θ, φ) e r .
(b) Verify that the density of free (E =
h̄2 k2
2m )
particle states for a particle of mass m in a box of size L3 is
ρdE =
mL3 k
dE .
2π 2 h̄2
(c) Work through the steps from Fermi’s Golden Rule to reproduce the f (θ, φ) we obtained in class for the
Yukawa potential.
3. Obtain the differential scattering cross section in the Born approximation for the potential V (r) = −V0 e−r/a
where V0 is a positive constant.
4. In the Born approximation, find the differential cross section for an electron scattering off a spherically symmetric
uniform charge distribution of radius R. Express your answer as a product of the Rutherford cross section for a
point charge and the square of a form factor F (q), where q is the magnitude of the momentum transfer. Find
an expression for F (q).
5. In class we will do s-wave scattering off a hard sphere. Now look at p-wave scattering off a hard sphere. Solve
the radial equation to find uk,1 (r), for r > r0 . Impose boundary conditions to find constants. Show that as
k → 0, δ1 ∼ (kr0 )3 . How does this compare to δ0 ?
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