Analytical study and exact solutions of the spin equation

Author(s)

Mário César Baldiotti

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2005-07-01

Examining board members

Dmitri Maximovitch Guitman; Bruto Max Pimentel Escobar; Josif Frenkel; Celso Luiz Lima; Jeferson de Lima Tomazelli

• Advisor: Dmitri Maximovitch Guitman

Abstract

The aim of the present work is to study in detail the so called spin equation, which can be used to describe the behavior of two-level systems. We recall that, for real external fields, this equation can be treated as a reduction of the Pauli equation to the 0+1 dimensional base. Initially, we present the relation between the spin equation and some other equations related to diferent physical problems. With these relations, we construct new solutions to the spin equation from the knowledge of the exact solutions of these other problems and, on the other hand, extend the applicability of the obtained solutions. After that, we describe the general solution of the spin equation, construct the evolution operator and solve the inverse problem, i.e., the construction of the external field from a given supposed solution. Finally, for the important case of real fields, we develop a method to construct new solutions from a previously known one, by the application of the so called Darboux transformation. In particular, we demonstrate the existence of Darboux intertwining operators which do not violate the specific structure of the two-level systems and allow the construction of external fields which are also given by real functions. As a result of all these developments, we present several new sets of exact solutions for the spin equation. (AU)