Time evolution, cyclic solutions and geometric phases for the generalized timedependent harmonic oscillator
Abstract
The generalized timedependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a cnumber vector which moves on a hyperboloid in a threedimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 January 2004
 DOI:
 10.1088/03054470/37/4/020
 arXiv:
 arXiv:quantph/0402159
 Bibcode:
 2004JPhA...37.1345L
 Keywords:

 Quantum Physics
 EPrint:
 revtex4, 28 pages, no figure