# Localized states of Dirac equation

## TL;DR

This paper extends the Dirac equation with a PT-symmetric momentum operator to produce localized wave packets as eigenstates, providing new solutions in 1D, 2D, and 3D that describe bound particles with specific angular momentum properties.

## Contribution

The paper introduces a novel PT-symmetric extension of the Dirac equation that yields localized eigenstates across different dimensions, expanding the understanding of Dirac particle localization.

## Key findings

- 1D solutions are bound particles with continuous energy spectrum.
- 2D solutions are localized wave packets with angular momentum eigenstates.
- 3D solutions are spherical localized wave packets with definite angular momentum.

## Abstract

In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.