Invariantes dinâmicos em mecânica quântica PT simétrica

Abstract

Non-Hermitian quantum systems have been used in many areas of the physics in last years, this show the importance these systems. In 1998, in a seminal paper, Bender has showed that such systems can be described by Hamiltonians non-Hermitians with parity and time reversal invariance. Since then the study of quantum systems described by Hamiltonians non-Hermitian have attracted the attention of many researchers. The great interest in this subject can be verified by the large quantity of publication in the literature. These papers mainly concerns to time-independent quantum systems. On the other hand, non-Hermitian time-dependent quantum systems don’t have been explored yet. However, we are interested in investigating time-dependent quantum systems modulated by nonHermitian Hamiltonians with PT symmetry. Among the ways of studying non-stationary systems our interest is, in particular, application of the dynamics invariant method that is explicitly time-dependent, proposed by Lewis and Riesenfeld, this method has been very successful in find a solution of the Schr¨odinger equation for non-stationary systems, obtaining exact and analytical solution. In this dissertation is discussed an extension of this method to solve the time-dependent Schro¨dinger equation described by non-Hermitian Hamiltonian with PT symmetry. As an application, we study the quantum motion of a particle subjected to a time-dependent non-Hermitian linear potential with PT symmetry. Thus, using a non-Hermitian linear invariant with PT symmetry we construct a solution type Gaussian package. Using this solution we calculate the position and momentum fluctuations and the corresponding uncertainty relation. We also show that the probability density is conserved.