Quantum-Corrected Statistical Method for Many-Particle Systems: The Density Matrix

Quantum corrections are obtained for the single-particle density matrix in a semiclassical ensemble where the distribution is unrestricted. A form for the density matrix containing explicitly a function of the wave-mechanical Hamiltonian operator is devised; a formalism is then developed to decompose this operator function into effectively classical and nonclassical parts. The classical part corresponds to the semiclassical density matrix and the quantum corrections are obtained from the nonclassical terms.The quantum-corrected density matrix for a spherically symmetrical and a one-dimensional system have been evaluated. The densities for a linear harmonic oscillator and a Coulomb potential for a Fermi-Dirac distribution have also been examined in some detail.