Principles of Quantum Mechanics · Problem 19.19

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · partial-wave-analysis

Problem

A particle of mass

m

and energy

E

scatters off a repulsive potential

V(r) = V_0 (a/r)^2

for

r < a

and

V = 0

for

r > a

, where

V_0 > 0

. (a) Show that only the

s

-wave contributes in the low-energy limit

ka \ll 1

. (b) Find the

s

-wave phase shift

\delta_0

for this potential. (c) Find the total cross section in the low-energy limit.