The most important consequence of the periodic potential is the formation of a small band gap at the boundary of the Brillouin zone.

This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.

These are the so called band-gaps, which can be shown to exist in any shape of periodic potential (not just delta or square barriers).

In a metal or semiconductor, the effect of the ions' periodic potential must be taken into account.

The resulting periodic potential may trap neutral atoms via the Stark shift.

The periodic potential of the lattice in this free electron model must be weak because otherwise the electrons wouldn't be free.

The scattering process results in the well known Bragg reflections of electrons by the periodic potential of the crystal structure.

Hill's equation is very general, as the θ-related terms may be viewed as a Fourier series expansion of a periodic potential.

Wannier functions have been extended to nearly periodic potentials as well.

The nearly free electron model rewrites the Schrödinger equation for the case of a periodic potential.