Assume that we have a wave function ψ which is an eigenstate of all the translation operators. As a special case of this, ψ ψ ( r + a j ) = C j ψ ψ ( r ) for j = 1, 2, 3, where C j are three numbers (the eigenvalues) which do not depend on r. It is helpful to write the numbers C j in a different form, by choosing three numbers θ 1, θ 2, θ 3 with e = C j : ψ ψ ( r + a j ) = e 2 π π i θ θ j ψ ψ ( r ) Again, the θ j are three numbers which do not depend on r. Define k = θ 1 b 1 + θ 2 b 2 + θ 3 b 3, where b j are the reciprocal lattice vectors (see above). Finally, define u ( r ) = e − − i k ⋅ ⋅ r ψ ψ ( r ) . Then u ( r + a j ) = e − − i k ⋅ ⋅ ( r + a j ) ψ ψ ( r + a j ) = ( e − − i k ⋅ ⋅ r e − − i k ⋅ ⋅ a j ) ( e 2 π π i θ θ j ψ ψ ( r ) ) = e − − i k ⋅ ⋅ r e − − 2 π π i θ θ j e 2 π π i θ θ j ψ ψ ( r ) = u ( r ) . This proves that u has the periodicity of the lattice. Since ψ ψ ( r ) = e i k ⋅ ⋅ r u ( r ) , that proves that the state is a Bloch state.