Symmetric Potential: Reasons for Eigenstate Solutions

[jostpuur]

I never learned this in the lectures (maybe I was sleeping), but now I think I finally realized what is the reason that eigenstate solutions of SE with a symmetric potential are either symmetric or antisymmetric. Is the argument this:

"The Hamiltonian and the space reflection operator commute, therefore they have common eigenstates" ?

If it is this, can somebody explain me why does a constant potential (which is also symmetric) have plane wave solutions, that are not symmetric or antisymmetric.

 

[Nesk]

Any quantum particle in a constant potential is in an un-bound state, i.e. a plain plane wave.