quantum mechanics - Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more complicated operators, like the Hamiltonians of the Hydrogen atom or harmonic oscillator?

Answer

The Wigner-Weyl transform of a function $f(x,p)$ is given by,

$$\Phi[f]= \frac{1}{4\pi^2}\iiiint f(x,p) \exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db$$

As you suggested, let us take the Hamiltonian of the harmonic oscillator, i.e.

$$\Phi[H]=\frac{1}{8m\pi^2}\iiiint p^2\exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db \\ + \frac{m\omega^2}{8\pi^2}\iiiint x^2\exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db$$

We concern ourselves with the last integral, as they are more or less analogous. As the first integration is over $x$, we may applying integration by parts and ignore $p$:

$$\frac{m\omega^2}{8\pi^2}\iiint \frac{1}{a^3}e^{ia(X-x)+ib(P-p)}(ia^2 + 2ax-2i) \, dp \,da \, db$$

Integrating with respect to $p$ is trivial:

$$\frac{m\omega^2}{8\pi^2}\iint \frac{b}{a^3} e^{ia(X-x)+ib(P-p)}(ax^2 +2-i2ax) \, da \, db$$

With the help of Mathematica 9, we may express the subsequent integral over $a$ in terms of a polyanomial, and the exponential integral function:

$$\frac{m\omega^2}{8\pi^2}\int \, b e^{ib(P-p)} \, \left[ (X-x)((ix^2+4x)-2X)\mathrm{Ei}(ia(X-x)) \\ -\frac{1}{a}e^{ia(X-x)}(i(X-x)+(x^2-i2x)+1) \right] \, db$$

The integral over $b$ is also trivial, as the integrand only features $b$ in the form $be^{b\dots}$ Hence,

$$-\frac{m\omega^2}{8\pi^2(P-p)^2} \left[ (X-x)((ix^2+4x)-2X)\mathrm{Ei}(ia(X-x)) -\frac{1}{a}e^{ia(X-x)}(i(X-x)+(x^2-i2x)+1) \right]e^{ia(X-x)+ib(P-p)}\left( ib(P-p)-1\right)$$

Apply the same procedure to the original first integral, combine the two, etc.