Parity Operator and odd potential function.

[sudipmaity]

Homework Statement

This is a university annual exam question: Show that for a potential V (-r)=-V (r) the wave function is either even or odd parity.

Homework Equations

The Attempt at a Solution

We can determine whether a wavefunctions' parity is time independent based on if the parity operator commutes with the hamiltonian.As far as I know for even parity potential V (-r)=V (r), hamiltonian and parity will commute and we can show that wave function to have even or odd parity.But for odd parity potential as given by the question the hamiltonian won't commute with the parity as it no longer remains invariant under parity operation.So wavefuntion should not have even or odd parity.I talked with some friends and some of them think that there might be some printing mistake in the question. What do you guys have to say.

 

[TSny]

To help see if the problem might have a typographical error, consider the one dimensional potential V(x) shown below. Note, V(-x) = -V(x).

Imagine a particle "trapped" in the region -a < x < 0. Consider roughly what the wavefunction for the ground state would look like. Would the wavefunction be even, odd, or neither?