How about ψ=ψe+ψo (where, of course, ψ must statisfy Schroedinger equation), and then prove that ψe≠0 and ψo≠0 cannot be simultaneously true.
Emm.....By wave function I assume you're talking about eigenfunction. Even though I can only prove that we can always find even solutions and odd solutions. As for what you said "prove wave function can be only even or odd." I really have no idea how to do it. I tried turin's method but didn't manage to get the desired answer. Let's wait for other people's opinion.
No, I'm pretty sure he needs wavefunctions and not eigenfunctions
woodywood, you may want to consider the parity operator:
and apply it to both the [itex]H\Psi[/itex] and [itex]\Psi[/itex] (after which you take the Hamiltonian of this latter one too--you should have [itex]PH\Psi[/itex] & [itex]HP\Psi[/itex]) then see how the two relate.
I'm pretty sure the OP needs eigenfunctions, not just wavefunctions in general. In fact, I'm pretty sure it is even more restricted to stationary eigenstates, because it is trivial to construct, by superposition, a general wavefunction that lacks definite parity, from any spectrum that includes both even and odd states.
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