The quantum Hamilton–Jacobi equations for the Dirac and the Dirac–Klein–Gordon systems

Abstract

For one dimensional systems, we derive from the Dirac spinor equation (DSE) the relativistic quantum stationary Hamilton-Jacobi equation (RQSHJES) for particles of spin 1 2 . We establish that the RQSHJES ± 1/2 is constituted of two equations, each one corresponding to the positive and negative energies (particle and antiparticle). The solution of the RQSHJES ± 1/2 is well established. We also investigate the Dirac-Klein-Gordon equation and set its corresponding Hamilton-Jacobi equation.