Eur. Phys. J. D 23, 305-313 (2003)
https://doi.org/10.1140/epjd/e2003-00043-7

Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

¹ State Key Laboratory of Modern Optical Instrumentation, Center for Optical and Electromagnetic Research, Zhejiang University, Hangzhou 310027, P.R. China
² Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China

Corresponding authors: ^a jqshen@coer.zju.edu.cn - ^b zhy@coer.zju.edu.cn

Received: 6 July 2002
Revised: 21 October 2002
Published online: 11 February 2003

Abstract

There exist a number of typical and interesting systems and/or models, which possess three-generator Lie-algebraic structure, in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl(2,C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of both the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schrödinger equations governing various three-generator Lie-algebraic quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, it is shown that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors and its adiabatic limit in time-dependent systems is briefly discussed.