I've noticed that for many quantum potentials, (the harmonic oscillator, the infinite square well, and the delta potential) the wavefunctions $\psi_n$ of stationary states are always real valued functions.
I couldn't explain this with observable-based arguments since only the probability distribution $\psi_n \psi_n^*$ is observable, which leaves open the possibility of complex wave-functions which nevertheless have real probability distributions.
So, is it a general principle that the wavefunctions associated with stationary states are real-valued? If so, why? If not, are there any simple examples of potentials where this does not hold?
