Wigner's quantum distribution function in spherical coordinates

N. A. Denisova

Teoretičeskaâ i matematičeskaâ fizika, Tome 27 (1976) no. 3, pp. 418-424

Résumé

The possibility is examined of applying the general method for introducing quantum distribution functions (QDF) proposed by Moyal to the concrete system of dynamical spherical variables $\widehat{\varphi}$, $\widehat{\theta}$, $\widehat r$, $\widehat p_{\varphi}$, $\widehat p_{\theta}$, $\widehat p_r$. The Wigner quantum distribution fimction found in this way in spherical coordinates determines the state of a system of particles only if the system is spherically or cylindrically symmetric. The changes are pointed out which must be made in Moyal's expressions for the basis operators $\widehat{\varphi}$, $\widehat{\theta}$, $\widehat r$, $\widehat p_{\varphi}$, $\widehat p_{\theta}$, $\widehat p_r$ if the distribution in the phase space $\varphi$, $\theta$, $r$, $n\hbar$, $m\hbar$, $p_r$ is to describe the state of any system of particles.