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Physical Review A

Method for the construction of the Lewis-Riesenfeld time-dependent invariants and their eigenvalue equations

M. A. de Ponte, Pedro M. Cônsoli, M. H. Y. Moussa

Published Sep 4, 2018• 16 citations
Publisher

Abstract

We propose here a method for the construction of the Lewis and Riesenfeld time-dependent invariants. Instead of directly invoking the equation that defines the invariant associated with a time-dependent Hamiltonian, our method allows the step-by-step construction of the invariant as a product of operators, and mainly a straightforward derivation of its eigenvalue equation.

DOI
10.1103/physreva.98.032102

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We propose here a method for the construction of the Lewis and Riesenfeld time-dependent invariants. Instead of directly invoking the equation that defines the invariant associated with a time-dependent Hamiltonian, our method allows the step-by-step construction of the invariant as a product of operators, and mainly a straightforward derivation of its eigenvalue equation.
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
This paper is concerned with the connection between density matrix method, supersymmetric quantum mechanics and Lewis-Riesenfeld invariant theory. It is shown that these three formulations share the common mathematical structure: specifically, all of them have the invariant operators which satisfies the Liouville-Von Neumann equation and the solutions to the time-dependent Schrödinger equation and/or Schrödinger eigenvalue equation can be constructed in terms of the eigenstates of the invariants.
We discuss the extension of the Lewis-Riesenfeld technique of solving the time-dependent Schr\"odinger equation to cases where the invariant has continuous eigenvalues. The extension is carried out for a general Ermakov system. We also discuss extensions to NNN-dimensional systems and the calculation of the propagator for a general Ermakov system.
Abstract We explore the possibility of modifying the Lewis–Riesenfeld method of invariants developed originally to find exact solutions for time-dependent quantum mechanical systems for the situation in which an exact invariant can be constructed, but the subsequently resulting time-independent eigenvalue system is not solvable exactly. We propose to carry out this step in an approximate fashion, such as employing standard time-independent perturbation theory or the WKB approximation, and subsequently feeding the resulting approximated expressions back into the time-dependent scheme. We illustrate the quality of this approach by contrasting an exactly solvable solution to one obtained with a perturbatively carried out second step for two types of explicitly time-dependent optical potentials.