There is another special case where the Hamiltonian may be translationally invariant. This type of translational symmetry is observed whenever the potential is periodic : V ( r j ± ± a ) = V ( r j ) In general, the Hamiltonian is not invariant under any translation represented by T ^ ^ j ( x j ) with x j arbitrary, where T ^ ^ j ( x j ) has the property: T ^ ^ j ( x j ) | r j ⟩ ⟩ = | r j + x j ⟩ ⟩ and, ( T ^ ^ j ( x j ) ) † † r ^ ^ j T ^ ^ j ( x j ) = r ^ ^ j + x j I ^ ^ (where I ^ ^ is the identity operator ; see proof above).