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Physical Review Letters

Foundations of the Determinantal Formalism in Time-Dependent Quantum Mechanics

André Fortini

Published Sep 17, 1984• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.53.1125

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

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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 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.
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.
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.
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.
The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.
Abstract The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.