Pseudo-invariant approach for a particle in a complex time-dependent linear potential

TL;DR

This paper introduces a pseudo-invariant approach to solve the Schrödinger equation for a particle in a complex, time-dependent linear potential, providing analytical solutions and addressing issues in previous methods.

Contribution

It proposes an alternative method using a linear pseudo-Hermitian invariant operator, improving the physical consistency of solutions compared to prior approaches.

Findings

Derived analytical Gaussian wave packet solutions.

Ensured correct normalization of invariant eigenfunctions.

Described classical motion with complex expectation values.

Abstract

The Lewis and Riesenfeld method has been investigated, by Ramos et al in Ref.[1], for quantum systems governed by time-dependent PT symmetric Hamiltonians and particularly where the quantum system is a particle submitted to action of a complex time-dependent linear potential. We discuss the method they used and propose an alternative one which leads to physically acceptable uncertainty product and to complex x and p expectation values but describe the classical motion. We used, for this situation, a linear pseudo hermitian invariant operator which allow us to solve analytically the time-dependent Schr\"odinger equation for this problem and to construct a Gaussian wave packet solution. The normalization condition for the invariant eigenfunctions with the Dirac delta function is correctly obtained, contrary to what is stated in Ref.[1].