Expectation values as a phase space average of Wigner functions

[Gabriel Maia]

Hi. I'm trying to prove that

$$[\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p)$$

where

$$\rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar})$$
is the Wigner function, being \rho a density matrix. On the other hand
$$\Omega_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\Omega|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar})$$
is the Wigner representation of the operator I'm interested in. The expectation value of an operator can be calculated from

$$[\Omega] = Tr(\rho\Omega) = \int dp \, \langle p|\rho\Omega|p\rangle$$
$$= \int dp^{\prime} \int dp \, \langle p|\rho|p^{\prime}\rangle\langle p^{\prime}|\Omega|p\rangle$$

Now, the matrix elements $$\langle p | \rho | p^{\prime} \rangle \,\,\, \mathrm{and} \,\,\, \langle p | \Omega| p^{\prime} \rangle$$

Should lead to the Wigner function and the Wigner representation of the operator but they only do so if
$$p=p^{\prime}$$

Do you know how can I solve this?

Thank you.

 

[member 553083]

The solution to this problem is to use a Fourier transform of the matrix elements. That is, we can write\langle p | \rho | p^{\prime} \rangle = \frac{1}{2\pi\hbar}\int dy \, \langle p-\frac{y}{2}|\rho|p+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar})and \langle p | \Omega | p^{\prime} \rangle = \frac{1}{2\pi\hbar}\int dy \, \langle p-\frac{y}{2}|\Omega|p+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}).Substituting these expressions into the expectation value equation yields[\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p),as desired.