The problem is related to the Hermite polynomials and operators of creation and annihilation. The creation operator:
a^{\dagger}\equiv \frac{1}{\sqrt{2}}\left(x- \frac{d}{dx}\right)
I played with the Rodrigues formula for Hermite poyinomials (for n=1?):
H_n (x)=(-1)^n e^{x^2}...