All Hermitian Hamiltonians have parity

It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator having the following properties: (i) is linear and Hermitian; (ii) commutes with H; (iii) ² = 1; (iv) the nth eigenstate of H is also an eigenstate of with eigenvalue (−1)ⁿ. Given these properties, it is appropriate to refer to as the parity operator and to say that H has parity symmetry, even though may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = p² + V(x), where V(x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has symmetry. This shows that symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = p² + V(x) have symmetry, but not all -symmetric Hamiltonians of this form are Hermitian.