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Physics LibreTexts

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 (Φ=e±ikx). If there is a weak periodic potential, Vcosax, to evaluate the energy shift to first order in degenerate perturbation theory the relevant matrix elements are:

e±ikxVcosaxeikxdx=Vcosaxdx=0;e±ikxVcosaxe±ikxdx=Vcosaxcos2kxdx

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.

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