Expectation values as a phase space average of Wigner functions

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1. Apr 12, 2017

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.

2. Apr 17, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.