But, whenever x j coincides with the period of the potential a , ( T ^ ^ j ( a ) ) † † V ( r ^ ^ j ) T ^ ^ j ( a ) = V ( r ^ ^ j + a I ^ ^ ) = V ( r ^ ^ j ) Since the kinetic energy part of the Hamiltonian H ^ ^ is already invariant under any arbitrary translation, being a function of p ^ ^ , the entire Hamiltonian satisfies, ( T ^ ^ j ( a ) ) † † H ^ ^ T ^ ^ j ( a ) = H ^ ^ Now, the Hamiltonian commutes with translation operator, i.e. they can be simultaneously diagonalised. Therefore, the Hamiltonian is invariant under such translation (which no longer remains continuous). The translation becomes discrete with the period of the potential.