Find wavefunction of harmonic oscillator

[BruceW]

ah right. I understand you now. If we make the total wavefunction $\psi$ either symmetric or anti-symmetric, then we are guaranteed that $\langle x \rangle$ is zero. And we know the mod square of the energy eigenstates are symmetric. Therefore, we can guarantee $\langle x \rangle = 0$ by getting rid of the cross terms.