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The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

For a lattice of period $a$, wavevectors differing by the reciprocal-lattice vector $2\pi /a$ describe the same translation eigenvalue. The first Brillouin zone may therefore be chosen as

Bloch theorem states that every energy eigenstate in a periodic potential may be chosen in the formTo prove it, let $T_a$ be translation by one period:Periodicity gives $[T_a,H]=0$. The translation operator is unitary, so it can be diagonalized within each energy eigenspace. Choose a common eigenstate,Unitarity gives $|\lambda |=1$, so $\lambda =e^{ika}$ for some real $k$. Thus $\psi (x+a)=e^{ika}\psi (x)$. Defining $u_k(x)=e^{-ikx}\psi (x)$ giveswhich proves the theorem.