Introduction to Quantum Mechanics · Problem 10.5

Scattering — David J. Griffiths

Physics — Quantum Mechanics · phase-shifts-finite-well

Problem

A particle of mass

m

and energy

E

is incident from the left on the potential

V(x) = \begin{cases} 0, & (x < -a), \\ -V_0, & (-a \leq x \leq 0), \\ \infty, & (x > 0). \end{cases}

(a) If the incoming wave is

Ae^{ikx}

(where

k = \sqrt{2mE}/\hbar

), find the reflected wave. Answer:

Ae^{-2ika}\left[\frac{k - ik'\cot(k'a)}{k + ik'\cot(k'a)}\right]e^{-ikx}, \text{ where } k' = \sqrt{2m(E+V_0)}/\hbar.

(b) Confirm that the reflected wave has the same amplitude as the incident wave. (c) Find the phase shift

\delta

(Equation 10.40) for a very deep well (

E \ll V_0

). Answer:

\delta = -ka