A Modern Approach to Quantum Mechanics · Problem 13.10

Photons and Atoms — John S. Townsend

Physics — Quantum Mechanics · p-wave-phase-shift-hard-sphere

Problem

Evaluate the P-wave phase shift

\delta_1

for scattering from a hard sphere, for which the potential energy is given by

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

Express your result in terms of

j_1(ka)

and

\eta_1(ka)

. Use the leading behavior of

j_1(\rho)

and

\eta_1(\rho)

for small

\rho

to show that

\delta_1 \to -(ka)^3/3

ka\to 0

, and thus

\delta_1

can indeed be neglected in comparison to

\delta_0

[see (13.81)] at sufficiently low energy.