# 13.5: Scattering of identical free particles with a periodic potential

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For a free particle moving in a 1D region of space there are two degenerate wavefunctions $$(\Phi = e^{\pm ikx})$$. If there is a weak periodic potential, $$V \cos ax$$, to evaluate the energy shift to first order in degenerate perturbation theory the relevant matrix elements are:

$\int e^{\pm i k x} V \cos a x e^{\mp i k x} d x=\int V \cos a x d x=0 ; \quad \int e^{\pm i k x} V \cos a x e^{\pm i k x} d x=\int V \cos a x \cos 2 k x d x \nonumber$

The second term is also zero, except in the case $$2k = a$$. This gives rise to the remarkable result: To first order, free particles are unaffected by a periodic potential unless it has half the wavelength. This is the basis of Bragg’s Law, x-ray and neutron diffraction.

This page titled 13.5: Scattering of identical free particles with a periodic potential is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.