Quantum SHO Wave Functions not Complex?

[LarryS]

The Hermite Polynomials are solutions to the Schrödinger equation for the Quantum Simple Harmonic Oscillator. But the Hermite Polynomials are real, not complex. I thought that solutions to the Schrödinger equation always had to be complex. What am I not understanding? Thanks in advance.

 

[jtbell]

You're missing the time-dependent phase factor. Energy eigenstates generally look like this, in their time-dependent form:

$$\Psi_n(x,t) = \psi_n(x) e^{-iE_n t / \hbar}$$