We can generalize this for 3 dimensions χ χ k 1 ( n 1 , a 1 ) χ χ k 2 ( n 2 , a 2 ) χ χ k 3 ( n 3 , a 3 ) = e i k ⋅ ⋅ τ τ and the generic formula for the wave function becomes: P ^ ^ R ψ ψ j = ∑ ∑ α α ψ ψ α α χ χ α α j ( R ) i.e. specializing it for a translation P ^ ^ ε ε | τ τ ψ ψ ( r ) = ψ ψ ( r ) e i k ⋅ ⋅ τ τ = ψ ψ ( r + τ τ ) and we have proven Bloch’s theorem.