The Relativistic Quantum Stationary Hamilton Jacobi Equation for Particle with Spin 1/2

TL;DR

This paper derives a relativistic quantum Hamilton-Jacobi equation for spin-1/2 particles from the Dirac equation in one dimension and provides explicit solutions for the two spin projections.

Contribution

It introduces a novel form of the quantum stationary Hamilton-Jacobi equation specifically for spin-1/2 particles, including explicit solutions for both spin projections.

Findings

Derived the quantum stationary Hamilton-Jacobi equation from Dirac's equation for spin-1/2 particles.

Presented explicit solutions for the two spin projection cases.

Demonstrated the equation's applicability in one-dimensional motion.

Abstract

For one dimensional motions, we derive from the Dirac Spinors Equation (DSE) the Quantum Stationary Hamilton-Jacobi Equation for particles with spin 1/2. Then, We give its solution. We demonstrate that the $QSHJES_{1\over2}$ have two explicit forms, which represent the two possible projection of the Spin 1/2.