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Journal of Physics A Mathematical and General

Determinantal solution of density matrix equations in time-dependent quantum mechanics. I. Constant perturbation

André Fortini

Published Dec 1, 1983• 6 citations
Publisher

Abstract

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

DOI
10.1088/0305-4470/16/17/015

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The general solution of the Schrödinger equation involving the time-dependent perturbation is presented in a compact and manageable form both for periodic and aperiodic perturbations. The method includes as a special case the solution of the Schrödinger equation involving the time-independent perturbation. Formulas ready for practical uses are explicitly described. The essential part of the procedure is the determinantal method in the Laplace-transformed space. The solution is applicable to strong perturbations as well as weak perturbations.

Journal of Mathematical Physics•Apr 1, 1966•2 citations
The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.
For pt.I see ibid. vol.16, p.3987-4000, (1983). The determinantal formalism for solving the time-dependent Schrodinger equation of the density matrix has been developed, in I, assuming a constant external perturbation. The important case of an harmonic perturbation is presented here. Detailed expressions are derived for diagonal and nondiagonal elements of the response, and the trace conservation theorem is shown to be satisfied at all times and any order. The results are again applied to the simple illustrative quantum system considered in I.
It is shown that the Fredholm-Laplace solution of the Schr\"odinger integral equation of the evolution operator in the cases of harmonic or constant external perturbations can be put into a physically meaningful determinantal quotient form. The solution exhibits the usual multiple transitions and corresponding linewidth upon expansion. The result lends itself fairly well to the setting up of correct tractable formulas in the important case of continuous spectra capable, for instance, of dealing with resonance phenomena. Furthermore, the determinantal form can be shown to obey the unitary requirement at all times and any order with respect to the applied field or the collision potential. An illustration is given in the simple model of a system submitted to a perturbation of constant matrix elements, in the spectral range of interest assumed of constant density of states.
It may look surprising that direct application of linear system theory to the Laplace transform of the time-dependent Schr\"odinger equation does not yield the evolution operator or density matrix in a physically reliable form, but requires a further "reduction procedure," earlier introduced in the frame of the so-called determinantal formalsim. It is shown that this procedure can in fact be viewed as a factorization of determinants intended so as to remove unexpected terms from the result.
The general solution of the Schrödinger equation involving the time-dependent perturbation is presented in a compact and manageable form both for periodic and aperiodic perturbations. The method includes as a special case the solution of the Schrödinger equation involving the time-independent perturbation. Formulas ready for practical uses are explicitly described. The essential part of the procedure is the determinantal method in the Laplace-transformed space. The solution is applicable to strong perturbations as well as weak perturbations.