A Modern Approach to Quantum Mechanics · Problem 13.12

Photons and Atoms — John S. Townsend

Physics — Quantum Mechanics · delta-shell-potential-phase-shift

Problem

Consider the spherically symmetric potential energy

\frac{2\mu V(r)}{\hbar^2} = \gamma\delta(r-a)

where

\gamma

is a constant and

\delta(r-a)

is a Dirac delta function that vanishes everywhere except on the spherical surface specified by

r = a

. (a) Show that the S-wave phase shift

\delta_0

for scattering from this potential satisfies the equation

\tan(ka + \delta_0) = \frac{\tan ka}{1 + (\gamma/k)\tan ka}

(b) Evaluate the phase shift in the low-energy limit and show that the total cross section for S-wave scattering is

\sigma \cong 4\pi a^2\left(\frac{\gamma a}{1+\gamma a}\right)^2