For each Bravais lattice vector R we define a translation operator T ^ ^ R which, when operating on any function f ( r ) shifts the argument by R : T ^ ^ R f ( r ) = f ( r + R ) Since all translations form an Abelian group, the result of applying two successive translations does not depend on the order in which they are applied, i.e. T ^ ^ R 1 T ^ ^ R 2 = T ^ ^ R 2 T ^ ^ R 1 = T ^ ^ R 1 + R 2 In addition, as the Hamiltonian is periodic, we have, T ^ ^ R H ^ ^ = H ^ ^ T ^ ^ R Hence, the T ^ ^ R for all Bravais lattice vectors R and the Hamiltonian H ^ ^ form a set of commutating operators. Therefore, the eigenstates of H ^ ^ can be chosen to be simultaneous eigenstates of all the T ^ ^ R : H ^ ^ ψ ψ = E ψ ψ T ^ ^ R ψ ψ = c ( R ) ψ ψ