Introduction to Quantum Mechanics · Problem 10.4

Scattering — David J. Griffiths

Physics — Quantum Mechanics · partial-wave-analysis-delta-function-shell

Problem

Consider the case of low-energy scattering from a spherical delta-function shell:

V(r) = \alpha\delta(r - a),

where

\alpha

and

a

are constants. Calculate the scattering amplitude,

f(\theta)

, the differential cross-section,

D(\theta)

, and the total cross-section,

\sigma

. Assume

ka \ll 1

, so that only the

\ell = 0

term contributes significantly. (To simplify matters, throw out all

\ell \neq 0

terms right from the start.) The main problem, of course, is to determine

C_0

. Express your answer in terms of the dimensionless quantity

\beta \equiv 2ma\alpha/\hbar^2

. Answer:

\sigma = 4\pi a^2 \beta^2/(1+\beta)^2