Introduction to Quantum Mechanics · Problem 10.7

Scattering — David J. Griffiths

Physics — Quantum Mechanics · phase-shifts-delta-function-shell

Problem

Find the

S

-wave (

\ell = 0

) partial wave phase shift

\delta_0(k)

for scattering from a delta-function shell (Problem 10.4). Assume that the radial wave function

u(r)

goes to 0 as

r \to 0

. Answer:

-\cot^{-1}\left[\cot(ka) + \frac{ka}{\beta\sin^2(ka)}\right], \quad \text{where} \quad \beta \equiv \frac{2m\alpha a}{\hbar^2}.