We define the translation operator T ^ ^ n ψ ψ ( r ) = ψ ψ ( r + T n ) = ψ ψ ( r + n 1 a 1 + n 2 a 2 + n 3 a 3 ) = ψ ψ ( r + A n ) with A = [ a 1 a 2 a 3 ] , n = ( n 1 n 2 n 3 ) We use the hypothesis of a mean periodic potential U ( x + T n ) = U ( x ) and the independent electron approximation with an Hamiltonian H ^ ^ = p ^ ^ 2 2 m + U ( x ) Given the Hamiltonian is invariant for translations it shall commute with the translation operator [ H ^ ^ , T ^ ^ n ] = 0 and the two operators shall have a common set of eigenfunctions. Therefore, we start to look at the eigen-functions of the translation operator: T ^ ^ n ψ ψ ( x ) = λ λ n ψ ψ ( x ) Given T ^ ^ n is an additive operator T ^ ^ n 1 T ^ ^ n 2 ψ ψ ( x ) = ψ ψ ( x + A n 1 + A n 2 ) = T ^ ^ n 1 + n 2 ψ ψ ( x ) If we substitute here the eigenvalue equation and dividing both sides for ψ ψ ( x ) we have λ λ n 1 λ λ n 2 = λ λ n 1 + n 2