Eur. Phys. J. Plus (2018) 133: 449
https://doi.org/10.1140/epjp/i2018-12251-3

Regular Article

Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having symmetry

¹ Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil
² Colégio Militar do Recife, Departamento de Ensino e Pesquisa do Exército Brasileiro, 50721-420, Recife, PE, Brazil

^* e-mail: iapedrosa@fisica.ufpb.br

Received: 19 July 2018
Accepted: 13 September 2018
Published online: 2 November 2018

Abstract

We discuss the extension of the Lewis and Riesenfeld invariant method to cases where the quantum systems are modulated by time-dependent non-Hermitian Hamiltonians having symmetry. As an explicit example of this extension, we study the quantum motion of a particle submitted to action of a complex time-dependent linear potential with symmetry. We solve the time-dependent Schrödinger equation for this problem and construct a Gaussian wave packet solution. Afterwards, we use this Gaussian packet to calculate the expectation values of the position and the momentum and the uncertainty product. We find that these expectation values are complex numbers and consequently the position and momentum operators are not observables.