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 $\exp(-2|\beta|^{2})$.

  • $\begingroup$ 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. $\endgroup$
    – Javier
    Mar 18, 2022 at 22:25
  • $\begingroup$ Oh I see. Let me look that up and try that. $\endgroup$
    – user135520
    Mar 19, 2022 at 1:02
  • $\begingroup$ Found this link: maths.dur.ac.uk/users/W.J.Zakrzewski/QM/gaussian.pdf will type up formal answer shortly $\endgroup$
    – user135520
    Mar 19, 2022 at 13:36


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