Is well know that if an Hamiltonian $\hat{H}$ commutes with the parity operator then exists a complete system of eigenstates with definite parity. So there will be even and odd states.
I noticed that every definite parity Hamilotnian I have worked with, has the ground state with the same parity of the Hamiltonian. Is this always true? How can I prove it? (maybe with the variational method?)
Morover I noticed the first excited states has always of opposite parity, the second excited state has the opposite and so on.. Is this also always true?
