One basic question

[mani5200]

Here is one of my test question. I did it wrong.

Specify whether the following statement is true or false and explain why? "The time independent Hamiltonian operator 'H' acting on an allowed state Psi will give the same state back, i.e "H Psi=E Psi, where E is the energy of the given state."

The answer is false. I don't know why but its false.

Can any one explain me this?

Thank you.

[newbee]

What does allowed state mean? If it means one of the possible physical states of the system and therefore superposition states are "allowed" then the answer is obvious.

[borgwal]

An "allowed state" means a "physical state", and that is in general any (normalized) superposition of eigenstates of the Hamiltonian. So, a general state is not necessarily an eigenstate of the Hamiltonian.

[Fredrik]

The answer should be "sometimes true, sometimes false" unless "allowed" means specifically that the state isn't an energy eigenstate.