3D Delta Function

Consider a particle of mass m which scatters in three dimension from a potential which is a shell at radius a:

(i)

Derive the exact s-wave expression for the scattering cross section in the limit of very low particle energy.

SOLUTION

For a particle of wave vector k, Schrodinger’s equation for the radial part of the wave function is

(1)

Only s-wave scattering is important at very low energies, so solve for ℓ = 0. Also define χ(r) = rR(r) and get

(2)

(3)

At r → 0, R(r) is well behaved, so χ = rR → 0. Thus we choose our wave functions to be

(4)

The quantity δ is the phase shift. We match the wave functions at r = a. The formula for matching the slopes is derived from (2):

(5)

Matching the function and slope produces the equations

(6)

(7)

which are solved to eliminate A and B and get

(8)

In the limit of low energy, we want ka → 0. We assume there are no bound states so that δ → kd, where d is a constant. We find in this limit:

(9)

(10)

We also give the formula for the cross section in terms of the scattering length d. The assumption of no bound state is that γa < 1.