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On the Widder inversion method in problems of statistical mechanics

А. В. Солдатов, JOSIP SEKE, Nikolay N. Bogolubov, V. A. Steklov

Published Jan 1, 2000• 0 citations
Publisher

Abstract

An alternative way of reconstructing a function from its Laplace transform \nusing the Widder inversion method was shown to be useful in treating some \nproblems of nonequilibrium statistical mechanics. As an example of a successful application of the method, a decay was investigated of the excited \natomic state in a simple, but nonetheless physically relevant, model featuring a two-level atom interacting with a continuum of field modes.

DOI
10.5488/cmp.3.2.371

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In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
A method for reconstructing the quantum state of a system of atoms or spins is proposed. The QQQ distribution is derived from the measured population in the ground state of a system that has interacted with an external field. Using multipole operators the QQQ function is inverted to derive the density matrix.
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.