Approximation scheme for the quantum Liouville equation using phase-space distribution functions

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Abstract
A general approximation scheme for solving the quantum Liouville equation has been developed and is presented here specifically within the Hartree approximation. This method uses the connection between the density matrix and the Wigner phase-space representations. By use of the Wigner representation the off-diagonal elements of the density matrix are expanded in powers of deviations from the mean momentum value. The time evolution of these deviations is governed by a closed self-consistent set of equations.