Introduction to Quantum Mechanics · Problem 10.17

Scattering — David J. Griffiths

Physics — Quantum Mechanics · born-approximation-one-dimensional

Problem

Use your result in Problem 10.16 to develop the Born approximation for one-dimensional scattering (on the interval

-\infty < x < \infty

, with no "brick wall" at the origin). That is, choose

\psi_0(x) = Ae^{ikx}

, and assume

\psi(x_0) \approx \psi_0(x_0)

to evaluate the integral. Show that the reflection coefficient takes the form:

R \approx \left(\frac{m}{\hbar^2 k}\right)^2 \left|\int_{-\infty}^{\infty} e^{2ikx}V(x)\,dx\right|^2. \tag{10.103}