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Journal of Physics A Mathematical and General

Analysis of quantum time-dependent evolution by the averaged variational principle

Mustapha Maamache, Hichem Choutri

Published Nov 6, 2001• 0 citations
Publisher

Abstract

In this paper we give a general method to solve a time-dependent Schrödinger equation. The formalism developed is based on the quantum variational principle conveniently `averaged'. In addition, we apply our method to study the time-dependent systems in which the Hamiltonians are linear functions of SU (1,1) and SU (2) Lie algebras.

DOI
10.1088/0305-4470/34/45/306

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In this paper we give a general method to solve a time-dependent Schrödinger equation. The formalism developed is based on the quantum variational principle conveniently `averaged'. In addition, we apply our method to study the time-dependent systems in which the Hamiltonians are linear functions of SU (1,1) and SU (2) Lie algebras.
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.
The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G 0 under under G , they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
A new approximation for solving the time-dependent Schrödinger equation is proposed. It improves the time-dependent Hartree approximation by including time-dependent unitary operators acting on the Hartree product. This allows for the approximate description of the correlation between the various degrees of freedom. The mathematical structure of the new approach is analyzed and an alternative Lie algebraic derivation is presented. By adopting two different time-dependent variational principles two different sets of equations are obtained. Differences between the two resulting methods are discussed.
We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.