Exact energy quantization condition for single Dirac particle in one-dimensional (scalar) potential well

TL;DR

This paper derives an exact quantization condition for the one-dimensional Dirac equation with a scalar potential well, enabling precise energy level calculations while maintaining physical symmetries and correct non-relativistic limits.

Contribution

It introduces a new exact energy quantization formula for the Dirac equation in scalar potential wells, extending non-relativistic results and ensuring physical consistency.

Findings

The formula accurately computes energy eigenvalues.

It generalizes non-relativistic quantization methods.

Numerical results confirm its effectiveness.

Abstract

We present an exact quantization condition for the time independent solutions (energy eigenstates) of the one-dimensional Dirac equation with a scalar potential well that gives only two `effective' turning points (defined by the roots of $V(x)+mc^2=\pm E$) for a given energy $E$ and satisfies $\min V(x)+mc^2\geq 0$. This result generalizes the previously known non-relativistic quantization formula and preserves many physically desirable symmetries, besides, attaining the correct non-relativistic limit. Numerical calculations demonstrate the utility of the formula for computing accurate energy eigenvalues.