CONDITION FOR A BIVARIATE NORMAL PROBABILITY-DISTRIBUTION IN PHASE-SPACE TO BE A QUANTUM STATE

In order that a bivariate normal probability distribution in phase space with variances sigma(q), sigma(p) and covariance sigma(q,p) may correspond to a Wigner distribution of a pure or a mixed state, it is n and sufficient that Heisenberg's uncertainty relation in Schrodinger form sigma(q)sigma(p)-sigma(q,p)2 greater-than-or-equal-to HBAR2/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.