Classical Mechanics (Kibble & Berkshire) · Problem 10.26

The angular momentum principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · quantum-scattering

Problem

Comparison with quantum scattering A uniform flux of particles is incident upon a fixed hard sphere of radius

a

. The particles that strike the sphere are reflected elastically. Show that the differential scattering cross section is

\sigma(\theta) = a^2/4

and that the total cross section is

S = \pi a^2

. The solution of the same problem given by quantum mechanics is

\sigma(\theta) = \frac{a^2}{(ka)^2}\left|\sum_{l=0}^{\infty}\frac{(2l+1)j_l(ka)P_l(\cos\theta)}{h_l(ka)}\right|^2, \quad S = \frac{4\pi a^2}{(ka)^2}\sum_{l=0}^{\infty}\left|\frac{(2l+1)j_l(ka)}{h_l(ka)}\right|^2,

where

P_l(z)

is the Legendre polynomial of degree

l

, and

j_l(z)

h_l(z)

are spherical Bessel functions order

l

. (Stay cool: these special functions should be available on your computer package.) The parameter

k

is related to the particle momentum

p

by the formula

p = \hbar k

, where

\hbar

is the modified Planck constant. When

ka

is large, one would expect the quantum mechanical values for

\sigma(\theta)

and

S

to approach the classical values. Calculate the quantum mechanical values numerically for

ka

up to about 30 (the calculation becomes increasingly difficult as

ka

increases), using about 100 terms of the series.