In quantum mechanics, the Hamiltonian represents the energy and dynamics of a system. Let | r T ⟩ ⟩ ≡ ≡ T ^ ^ | r ⟩ ⟩ be a newly translated state (the argument of T ^ ^ ( x ) is irrelevant here and is temporarily dropped for brevity). A Hamiltonian is said to be invariant if ⟨ ⟨ r T | H ^ ^ | r T ⟩ ⟩ = ⟨ ⟨ r | H ^ ^ | r ⟩ ⟩ or ⟨ ⟨ r | T ^ ^ − − 1 H ^ ^ T ^ ^ | r ⟩ ⟩ = ⟨ ⟨ r | H ^ ^ | r ⟩ ⟩ This implies that T ^ ^ − − 1 H ^ ^ T ^ ^ = H ^ ^ H ^ ^ T ^ ^ − − T ^ ^ H ^ ^ = [ H ^ ^ , T ^ ^ ] = 0