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DETERMINATION OF THE EIGEN-ENERGIES AND FUNCTIONS FROM THE MAXWELL- BOLTZMANN DENSITY OPERATOR. II. CASE OF PERTURBED SYSTEMS

Pieter Smet, Jacques Tillieu

Published Nov 1, 1962• 0 citations
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Abstract

Previously described methods for the determination of eigen-energies and corresponding projectors from the Maxwell-Boltzmann density operator are adapted to a perturbed system. As an example, the anharmonic oscillator problem is examined. (auth)

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UTILIZATION OF THE DENSITY OPERATOR FOR THE DETERMINATION OF THE EIGENENERGIES AND EIGENFUNCTIONS OF A CANONICAL GROUP

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Two methods are indicated for obtaining eigenenergies and corresponding projectors of a closed quantum system from the density operator describing this system in thermodynamic equilibrium and considered as a solution of Bloch's equation. The first method uses the limit of null temperatures; the second one, the properties of Laplace's transform. The harmonic oscillator is treated as example. (auth)

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We present here a succinct account of the finite-temperature generalisation of the coupled cluster (CC) method. It requires the concept of thermal normal ordered products whose Boltzmann trace is zero. The basic idea is to express the Boltzmann operator as a normal ordered exponential containing cluster operators and a number, and express the free energy as a logarithm of a suitable Boltzmann trace, where only the number part in the ordered exponential survives. The free energy is manifestly extensive. Free energies of Lipkin model and anharmonic oscillators obtained from the grand partition function are discussed as example applications.

International Journal of Modern Physics B•Nov 10, 2003•19 citations
The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.
We present here a succinct account of the finite-temperature generalisation of the coupled cluster (CC) method. It requires the concept of thermal normal ordered products whose Boltzmann trace is zero. The basic idea is to express the Boltzmann operator as a normal ordered exponential containing cluster operators and a number, and express the free energy as a logarithm of a suitable Boltzmann trace, where only the number part in the ordered exponential survives. The free energy is manifestly extensive. Free energies of Lipkin model and anharmonic oscillators obtained from the grand partition function are discussed as example applications.