13.5: Scattering of identical free particles with a periodic potential
( \newcommand{\kernel}{\mathrm{null}\,}\)
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±ikxVcosaxe∓ikxdx=∫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.