Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems
type: Item
instance of: scholarly article
title: Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems
zbMATH Open document ID: 1054.81017
author: Wensen Liu
published in: Annals of Physics
publication date: 2004-08-19T00:00:00Z
review text: The author investigates the time evolution of quantum systems which is described by the following time-dependent Hamiltonian: \[ H(t)=B_+(t)J_++B_-(t)J_-+B_0(t)J_0+B_1(t)e_1+B_2(t)e_2+B_3(t)e_3, \] where \(B_j(t), (j=\pm,0,1,2,3)\) are nonsingular functions of time, the \(J_j(j=\pm,0)\) and \(e_j(j=1,2,3)\) are the generators of the \(su(1,1)\) and Heisenberg-Weyl algebra \(h_3\), respectively. In section 2 he develops a time-depending closed-form formulation of the linear transformation for harmonic-oscillator annihilation and creation operators by means of a Lie algebraic approach. It is shown that the differential equation satisfied by the unitary operator performing the transformation possesses the two-photon Lie algebraic structure and reveals the interrelationship between the unitary operator and the transformation. In section 3, by combining the above formulation with the time evolution given the Hamiltonian, another method to solve the time evolution equation of time dependent quadratic Hamiltonian systems is established. In the last section 4 the developed methods are verified for the motion of a charged particle with variable mass in a time-dependent electric field.
reviewed by: Farruh Mukhamedov
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 2092927
zbMATH Keywords: unitary transformation
zbMATH Keywords: time evolution
MaRDI profile type: Publication
full work available at URL: j.aop.2004.04.002
OpenAlex ID: W2168569744
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DOI: 10.1016/J.AOP.2004.04.002
description: scientific article; zbMATH DE number 2092927