Time-Dependent Quantum Systems

Abstract

This thesis is devoted to the demonstration of the idea that the study of properties of a time-dependent quantum system should be based on the properties of its schrödinger operator S(t) = H(t) - ih∂/∂t rather then the Hamiltonian H(t).In chapter 1 an evolution of a. quantum-mechanical system under a, non-adiabatic external perturbation at the time interval [0,T] is considered. It is shown that all the cyclic states of the system are determined by the eigenvalues ε and associated eigenvectors of the restriction of its schrödinger operator, sq(t), to some Hilbert space. The set of linearly independent cyclic states possess some properties similar to the properties of the stationary states of a closed system. The Berry phase of a state associated with an eigenvector of the discrete spectrum of Sq(t), which is a single-valued function of M = 2F/T , is supplied by the partial derivative of the corresponding eigenvalue ε with respect to ω . The approach developed is illustrated by several examples. In chapter 2 general consequences of the existence of a unitary operator W(t) which transforms the schrödinger operator s o(t) of a quantum system into the Schrödinger operator S o(t) of another system, S(t) = W(t)S o(t)W+(t), are obtained. These results are illustrated by an alternative treatment of the time-dependent quantum oscillator which is presented in chapters 2 and 3. The treatment is based on exact time-dependent invariants and the family of unitary operators {W (t)} realizing the transformation of the Schrödinger operator of the time-dependent oscillator into the Schrödinger operator of the force-free oscillator with a constant frequency, and provides a parallel description of the two systems.