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Physica A Statistical Mechanics and its Applications

Density matrix of anharmonic oscillators by a thermodynamic variation–perturbation technique, a closed form approach

Wolfgang Witschel

Published Nov 1, 2001• 3 citations
Publisher
DOI
10.1016/s0378-4371(01)00332-6

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A thermodynamic perturbation method for the quantum-anharmonic oscillator is studied, based upon the Bloch equation. By this method the density matrix (Green's function) for the anharmonic oscillator can be obtained in a compact form for each order of the perturbation series. It is shown that the density matrix of the anharmonic oscillator can be readily expanded in series of the Hermite polynomials. The various expressions of the expansion series for the density matrix are also derived by a unified method.
Two series representations of the propagator associated with the quantum anharmonic-oscillator are developed. A closed form for the Dyson series expansion of the propagator is obtained by using a phase-space perturbation technique. For Abelian interactions the Dyson series can be rearranged into an exponentiated connected-graph series. This representation is structurally similar to the cluster expansions of propagators associated with perturbations of the kinetic-energy Hamiltonian. The connected-graph series is particularly useful for the development of semiclassical expansions such as the WKB expansion. The analytic structure in the physical parameters such as mass, time and Planck's constant is readily extracted from the connected-graph series because of the explicit closed-form formula for the series.