Principles of Quantum Mechanics · Problem 19.12

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · partial-wave-phase-shifts

Problem

Find the

s

-wave phase shift for scattering from a finite spherical well:

V(r) = \begin{cases} -V_0, & r < a \\ 0, & r > a \end{cases}

with

V_0 > 0

. Show that

\tan(\delta_0 + ka) = \frac{k}{\kappa}\tan(\kappa a)

where

\kappa = \sqrt{2m(E+V_0)}/\hbar

and

k = \sqrt{2mE}/\hbar

. Determine the condition for a zero-energy resonance (i.e.,

\delta_0 = \pi/2

k\to 0