# Average value of an operator on vacuum state [closed]

I'm trying to calculate $$<0|e^{a\hat{x}^2}e^{b\hat{x}}e^{c\hat{p}}|0>$$ where $a$, $b$ and $c$ are complex numbers, $\hat{x}$ is the position operator, $\hat{p}$ is momentum operator and $|0>$ refers to vacuum state (i.e. Fock state with 0 photons).

So far, I have tried direct expansion of exponential in Taylor series, playing with Baker-Campbell-Hausdorff sort of formulae, and trying to recognize known states, such as squuezed states in the formula I'm asking about. This efforts were unsuccessful, though. Any ideas?

Assuming that all quantities are written in units of the harmonic oscillator natural length $\sqrt{\hbar/m\omega}$, and using the ground-state wavefunction $\psi_0(x)=\frac{1}{\pi^{1/4}}e^{-x^2/2}$, one can perform the average by inserting enough resolution of the identity with respect to $x$ and $p$, to obtain (whether all constants are correct is left as an exercise to the reader) : $$\langle 0|e^{a\hat x^2}e^{b\hat x}e^{c\hat p}|0\rangle=\frac{e^{\frac{b^2+c^2-2 i b c-2 a c^2}{4(1-a)}}}{\sqrt{1-a}}.$$
Note that this average is valid only if $a<1$.