As above, let T ^ ^ n 1 , n 2 , n 3 denote a translation operator that shifts every wave function by the amount n 1 a 1 + n 2 a 2 + n 3 a 3, where n i are integers. Because the crystal has translational symmetry, this operator commutes with the Hamiltonian operator. Moreover, every such translation operator commutes with every other. Therefore, there is a simultaneous eigenbasis of the Hamiltonian operator and every possible T ^ ^ n 1 , n 2 , n 3 operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).