13.5: Scattering of identical free particles with a periodic potential

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.