# Difference between Wigner function in coherent space and coordinate space

Such a density operator $$\hat{\rho}$$ match with a Wigner function in coherent space: $$W(\alpha,\alpha^{*})$$ $$\alpha$$ and $$\alpha^{*}$$ are $$C-number$$ ( $$\alpha^{*}$$ denotes complex conjugate of $$\alpha$$) corresponding to annihilation and creation operator $$a$$ and $$a^\dagger$$.

Or we may write Wigner function under coordinate space： $$W(x,p)$$ where $$x$$ and $$p$$ are also $$C-number$$ corresponding to position and momentum operator $$\hat{x}$$ and $$\hat{p}$$.

My question is, if I have a Wigner function under coordinate space $$W(x,p)$$, may I obtain $$W(\alpha,\alpha^{*})$$ just simply get through: $$\alpha→x+p$$ $$\alpha^*→x-p$$ or vice versa: $$x=(\alpha+\alpha^{*})/2$$ $$p=(\alpha-\alpha^{*})/2i$$ Is there such a simple relationship between them?

• WP. Commented Mar 19, 2022 at 11:54

$$x=\sqrt{\frac{\hbar}{2m\omega}}(\alpha^* + \alpha),\, p=i\sqrt{\frac{\hbar m\omega}{2}}(\alpha^* - \alpha).$$
On a side note, the coherent representation is sometimes more useful because for example in circuit QED or cavity QED, since $$|\alpha|^2$$ is the average number of photons - this is immediately more useful since they can control "how many photons" to pump and explore or probe phase space experimentally.