Quantum corrections (within the classical path approximation) to the boltzmann density matrix

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A previously derived classical path approximation for diagonal matrix elements of the Boltzmann density matrix (i.e., the particle density) is expanded in a power series in about the classical density. Comparing term by term with Wigner's expansion of the exact quantum mechanical density, one sees that the first quantum correction to the classical density (which involves first and second derivatives of the potential energy) is given correctly by the classical path approximation. wigner's expression for the second correction term, however, involves the first four derivatives of the potential, and the classical path approximation gives only the part of this term that involves the first three derivatives.