Bloch theorem

In a crystalline solid, the potential experienced by an electron is periodic.

$$V(x) = V(x + a)$$

where $a$ is the crystal period/ lattice constant. In such a periodic potential, the one electron solution of the Schrödinger equation is given by the plane waves modulated by a function that has the same periodicity as that of the lattice:

$$\psi(x) = u_k(x) e^{\pm ikx}$$

Where, $u_k(x) = u_k(x + a)$

$$\psi(x + a) = e^{iKa} \psi(x)$$ $$|\psi(x+a)|^2 = |\psi(x)|^2$$

Applying boundary condition: $\psi(x + Na) = \psi(x)$, and that implies:

$$K = \frac{2\pi n}{Na}, (n = 0, \pm 1, \pm 2, \dots)$$

Such a periodic potential can be modelled by a Dirac comb (Dirac delta potential at each lattice point) or Kronig-Penney model where we have finite square well potential.