Question

A particle of mass m is scattered by a potential V(r) = V₀ exp(-r/a).

(a) Find the differential scattering cross section in the first Born approximation. Sketch the angular dependence for small and large k, where k is the wave number of the particle being scattered. At what k value does the scattering begin to be significantly non-isotropic? Compare this value with the one given by elementary arguments based on angular momentum.

(b) The criterion for the validity of the Born approximation is

\(|\Delta\psi^{(1)} (0) / \psi^{(0)} (0) |<<1, \)

where \(\Delta\psi^{(1)}\) is the first order correction to the incident plane wave \(\psi^{(0)}\). Evaluate this criterion explicitly for the present potential. What is the low-k limit of your result? Relate it to the strength of the attractive potential required for the existence of bound states (see the statement of problem). Is the high-k limit of the criterion less or more restrictive on the strength of the potential?

Answer 1

(a) The first Born approximation gives

where q = 2k sin (\(\theta\)/2), qh being the magnitude of the momentum transfer in the scattering. Hence

The angular distribution \(\sigma(\theta) /\sigma(0)\) is plotted in Fig. for ka = 0 and ka = 1.

It can be seen that for ka ≥ 1, the scattering is significantly non-isotropic. The angular momentum at which only s-wave scattering, which is isotropic, is important must satisfy

a . kh ≤ h, i.e., ka ≤ 1.

When ka \(\sim\) 1, the scattering begins to be significantly non-isotropic. This is in agreement with the result given by the first Born approximation.

(b) The wave function to the first order is

Hence

The criterion for the validity of the first Born approximation is then

In the low-k limit, ka << 1, the above becomes

In the high-k limit, ka >> 1, the criterion becomes

Since in this case k >> \(\frac 1a\) the restriction on |V₀| is less than for the low-k limit.