Riley hobson bence · Problem 4.36

Series and limits

Mathematics — Methods · lhopitals-rule-scattering-amplitude

Problem

In quantum theory, a certain method (the Born approximation) gives the (so-called) amplitude

f(\theta)

for the scattering of a particle of mass

m

through an angle

\theta

by a uniform potential well of depth

V_0

and radius

b

(i.e. the potential energy of the particle is

-V_0

within a sphere of radius

b

and zero elsewhere) as

f(\theta) = \frac{2mV_0}{\hbar^2 K^3}(\sin Kb - Kb\cos Kb).

Here

\hbar

is the Planck constant divided by

2\pi

, the energy of the particle is

\hbar^2 k^2/(2m)

and

K

2k\sin(\theta/2)

. Use l'H\^{o}pital's rule to evaluate the amplitude at low energies, i.e. when

k

and hence

K

tend to zero, and so determine the low-energy total cross-section. [Note: the differential cross-section is given by

|f(\theta)|^2

and the total cross-section by the integral of this over all solid angles, i.e.

2\pi \int_0^\pi |f(\theta)|^2 \sin\theta\, d\theta