# Heisenberg uncertainty relation for relativistic electrons

## TL;DR

This paper derives the Heisenberg uncertainty relation for relativistic electrons using the Dirac equation, revealing different bounds in nonrelativistic and ultrarelativistic limits, with implications for quantum uncertainty principles.

## Contribution

It provides a new derivation of the uncertainty relation for relativistic electrons and explores its behavior across different energy regimes.

## Key findings

- Nonrelativistic limit attains the standard lower bound of 3/2ħ.
- Ultrarelativistic limit yields the same bound as for photons.
- Wave functions achieving the bounds are singular in the nonrelativistic case.

## Abstract

The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound $3/2\hbar$ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for photons.