Semiclassical propagation in the coherent-state representation

Author(s)

Thiago de Freitas Viscondi

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Física Gleb Wataghin
• Defense date: 2013-02-18

Examining board members

Marcus Aloizio Martinez de Aguiar; Amir Ordacgi Caldeira; Eduardo Miranda; Alexandre Dias Ribeiro; Marcel Novaes

• Advisor: Marcus Aloizio Martinez de Aguiar

Abstract

The semiclassical propagation comprises the development and application of methods for obtaining approximate solutions to the time-dependent Schrödinger equation, assuming the hypothesis that the classical action of the system is much greater than the Planck constant. In this context, the quantum propagator represents an element of central interest, since this quantity corresponds to the probability amplitude for the transition between two states of thephysical system. In a preliminary stage of our work, we employ the generalized concept of coherent states to reformulate the quantum propagator in terms of a path integral. Then, with use of the saddlepoint method, we conduct a detailed derivation of the semiclassical approximation for the propagator corresponding to a wide class of dynamical groups. The application of this formal result depends on the resolution of classical equations of motion under boundary conditions, considering a phase space with doubled dimension. Generally, the search for classical trajectories subject to boundary values demonstrates high computational cost and technical complexity. For this reason, we have developed three distinct semiclassical approximations exclusively determined by initial conditions. In a first situation, we elaborate a propagation method composed of an integral over classical solutions in the doubled phase space. In the second case, with the formulation of the semiclassical time-evolution operator, we eliminate the need for the duplication of the classical degrees of freedom. The third approach, designated as corrected classical propagator, is defined by the contribution of a single trajectory. In order to evaluate the accuracy and efficiency of the obtained semiclassical expressions, we exemplify the application of these theoretical tools for the coherent states of SU(2) and SU(3). At last, we present an extensive discussion on the advantages introduced by the doubled phase space in implementing a semiclassical approximation. In this way, we find that tunneling classical solutions have an important participation in the accurate description of the partial penetration of a wave packet in a finite potential barrier. Furthermore, we show that the quantum phenomenon of reflection by an attractive potential is directly associated to the occurrence of trajectories with non-classical behavior. (AU)

• FAPESP's process: 08/09491-9 - Semiclassical limit of finite quantum systems
• Grantee: Thiago de Freitas Viscondi
• Support Opportunities: Scholarships in Brazil - Doctorate