# Computing $\frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha)\exp(-|\lambda|^{2}/2)d^{2}\lambda$

This came up when attempting to do a routine calculation of Wigner function of the vacuum state

$$\frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha-|\lambda|^{2}/2)d^{2}\lambda$$ I feel like there is a standard trick of completing the square, where we multiply by $$\exp(-|\alpha|^{2}/2)$$ in fact, I think the answer should be $$\exp(-|\alpha|^{2}/2)$$ by knowing that $$Q_{|0 \rangle \langle 0|}(\alpha)=\frac{1}{\pi}e^{-|\alpha|^{2}}$$ and that one can get the $$Q$$ function from the Wigner function by convolution with $$\exp(-2|\beta|^{2})$$.

• Do you know how to calculate an integral of the form $\int dx \exp(-ax^2+bx)$? Because you can just write $\lambda$ in terms of its real and imaginary parts and split it into two such integrals. Mar 18, 2022 at 22:25
• Oh I see. Let me look that up and try that. Mar 19, 2022 at 1:02
• Found this link: maths.dur.ac.uk/users/W.J.Zakrzewski/QM/gaussian.pdf will type up formal answer shortly Mar 19, 2022 at 13:36