Critical Points of a Wigner function

Question

I am interested in calculating the critical points of a Wigner function $$ W(x,p)=\frac{1}{\pi}\int_{-\infty}^\infty\left\langle x+y\middle|\rho\middle|x-y\right\rangle e^{-2ipy}\mathrm{d}y $$ Unfortunately, my knowledge of calculus and linear algebra is only rudimentary. I know that I can do this by calculating the first and second partial derivatives with respect to $x$ and $p$, but I have no idea how to evaluate $$ \frac{\partial}{\partial x} \left\langle x+y \middle |\rho \middle| x-y\right\rangle $$ In particular, I am not even sure how $\left|x\right\rangle$ should look in this expression for an arbitrary $\rho$. Does it depend on the basis in which $\rho$ is written?

Aside from that, I was wondering if there is a more straightforward (established maybe? I found nothing so far) way of calculating the critical points of a Wigner function.

Answer 1

I'm not sure why you might be interested in $W(x,p)$ as a plain function (which it is), since it is meant to be acted on by star products of functions of x and p, unless it is inside a phase-space integral. In any case, you of course know that $$ \frac{\partial}{\partial x} \left\langle x+y \middle |\hat \rho \middle| x-y\right\rangle = \left\langle x+y \middle |{i\over \hbar}[\hat p,\hat \rho ] \middle| x-y\right\rangle , $$ and it's trivial to derive the exponential inside the integral, $$ \frac{\partial}{\partial p}W(x,p)=\frac{-2i}{\pi}\int_{-\infty}^\infty\left\langle x+y\middle|\hat \rho\middle|x-y\right\rangle e^{-2ipy}y\mathrm{d}y . $$

So it very much depends on the nature of the (operator) density matrix. For example, for the ground state of the quantum harmonic oscillator, a plain phase-space Gaussian when nondimensionalized, m=1, ω=1, $$ W(x,p)={1\over \pi \hbar} e^{-{x^2+p^2 \over 2\hbar}}, $$ with trivial critical structure as a plain function in phase space.