Quantum speed limit for a relativistic electron in a uniform magnetic field

TL;DR

This paper investigates the quantum speed limit for a relativistic electron in a magnetic field, highlighting differences between relativistic and non-relativistic models and their implications for electron dynamics.

Contribution

It provides a detailed analysis of the quantum speed limit considering relativistic effects and compares Dirac and Schrödinger-Pauli equations under various magnetic field strengths.

Findings

Relativistic effects significantly alter the quantum speed limit in strong magnetic fields.

Schrödinger-Pauli equation predicts superluminal electron speeds in extreme conditions.

Relativistic and non-relativistic dynamics diverge notably in high magnetic field regimes.

Abstract

We analyze the influence of relativistic effects on the minimum evolution time between two orthogonal states of a quantum system. Defining the initial state as an homogeneous superposition between two Hamiltonian eigenstates of an electron in a uniform magnetic field, we obtain a relation between the minimum evolution time and the displacement of the mean radial position of the electron wavepacket. The quantum speed limit time is calculated for an electron dynamics described by Dirac and Schroedinger-Pauli equations considering different parameters, such as the strength of magnetic field and the linear momentum of the electron in the axial direction. We highlight that when the electron undergoes a region with extremely strong magnetic field the relativistic and non-relativistic dynamics differ substantially, so that the description given by Schroedinger-Pauli equation enables the electron traveling faster than c, which is prohibited by Einstein's theory of relativity. This approach allows a connection between the abstract Hilbert space and the space-time coordinates, besides the identification of the most appropriate quantum dynamics used to describe the electron motion.