$\mathcal{PT}$-Symmetric Generalized Extended Momentum Operator

TL;DR

This paper introduces a $ ext{PT}$-symmetric generalized extended momentum operator (GEMO), demonstrating its properties, associated Hamiltonian, and exact energy spectrum solutions for a quantum particle, expanding the framework of non-Hermitian quantum mechanics.

Contribution

It proposes a new $ ext{PT}$-symmetric version of the GEMO, showing its compatibility with the extended uncertainty principle and deriving exact energy spectra for specific quantum systems.

Findings

The $ ext{PT}$-symmetric GEMO satisfies the extended uncertainty principle.

The associated Hamiltonian remains non-Hermitian but $ ext{PT}$-symmetric with real eigenvalues.

Exact solutions for the energy spectrum of a quasi-free particle are obtained.

Abstract

We develop further the concept of generalized extended momentum operator (GEMO), which has been introduced very recently in \citep{M.H2}, and propose the so called $\mathcal{PT}$-symmetric GEMO. In analogy with GEMO, the $\mathcal{PT}$-symmetric GEMO also satisfies the extended uncertainty principle (EUP) relation. Moreover, the corresponding Hamiltonian that is constructed upon the $\mathcal{PT}$-symmetric GEMO, with a real or $\mathcal{PT}$-symmetric potential, remains non-Hermitian but $\mathcal{PT}$-symmetric and consequently its energy and momentum eigenvalues are real. We apply our formalism to a quasi-free quantum particle and the exact solutions for the energy spectrum are presented.