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

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 Schrödinger 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].

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