For two wave vectors and
the solutions of Schrödinger equation (2.68) are related to each other as
. This leads to equal eigenvalues
and equal wave functions
It can be seen that each energy branch has the same period as the reciprocal lattice. As the functions
are periodic, they have
maxima and minima which determine the width of the bands.
It should be noted that the wave vector in (2.66) can always be chosen in a way to belong to the first Brillouin zone because any vector
out of the first Brillouin zone can be represented as the sum
, where is a vector of the reciprocal
lattice. Using the equivalent form of Bloch's theorem:
together with (2.71) and the equality
one obtains (2.72) for vector .