[Paper Review] The Relativistic Quantum Stationary Hamilton Jacobi Equation for Particle with Spin 1/2

T. Djama|ArXiv.org|Nov 10, 2003

Quantum optics and atomic interactions5 references3 citations

TL;DR

This paper derives the relativistic quantum stationary Hamilton-Jacobi equation (RQSHJE) for spin-1/2 particles in one dimension from the Dirac equation, demonstrating two distinct forms corresponding to the two spin projections ($m_s = \pm 1/2$). It establishes a deterministic framework for relativistic spin-1/2 particles by solving the RQSHJE using wavefunction decomposition and Wronskian techniques, showing that the equations reduce to the standard QSHJE in the non-relativistic limit and to the classical Hamilton-Jacobi equation in the $\hbar \to 0$ limit.

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.

Motivation & Objective

• To extend the deterministic quantum formalism, previously developed for spinless particles, to relativistic spin-1/2 systems.
• To derive the relativistic quantum stationary Hamilton-Jacobi equation (RQSHJE) for spin-1/2 particles from the Dirac equation in one dimension.
• To demonstrate that the RQSHJE for spin-1/2 has two explicit forms, each corresponding to one of the two possible spin projections ($m_s = +1/2$ and $m_s = -1/2$).
• To establish a solution method for the RQSHJE using wavefunction decomposition and Wronskian analysis.
• To show that the RQSHJE reduces to the standard non-relativistic QSHJE and classical Hamilton-Jacobi equation in appropriate limits.

Proposed method

• Derives the RQSHJE for spin-1/2 by decomposing the Dirac spinor wavefunction into real amplitude and phase functions, using $\theta(x) = A(x)(\alpha_+ e^{iS_0/\hbar} + \alpha_- e^{-iS_0/\hbar})$ and $\phi(x) = B(x)(\beta_+ e^{iZ_0/\hbar} + \beta_- e^{-iZ_0/\hbar})$.
• Substitutes the wavefunction ansatz into the one-dimensional Dirac equation, leading to second-order differential equations for $\theta$ and $\phi$.
• Applies the Wronskian method to the solutions of the $\theta$-equation, using $W = \theta \frac{d\theta_2}{dx} - \theta_2 \frac{d\theta}{dx} = \alpha (E - V + m_0c^2)$ to derive the RQSHJE for $S_0$.
• Uses the Schwarzian derivative to express quantum corrections in the RQSHJE, leading to terms involving $\{S_0, x\}$ and derivatives of $E - V + m_0c^2$.
• Derives a second RQSHJE for $Z_0$ by analogous methods, corresponding to the other spin projection.
• Confirms consistency by showing that the derived RQSHJE is satisfied when the wavefunction ansatz is substituted back into the original equations.

Experimental results

Research questions

• RQ1 How can the relativistic quantum stationary Hamilton-Jacobi equation be derived for spin-1/2 particles from the Dirac equation in one dimension?
• RQ2 What is the mathematical structure of the RQSHJE for spin-1/2, and does it admit multiple solutions corresponding to spin projections?
• RQ3 How do the two forms of the RQSHJE for $m_s = \pm 1/2$ relate to the non-relativistic and classical limits?
• RQ4 Can the RQSHJE for spin-1/2 be solved explicitly using wavefunction decomposition and Wronskian techniques?
• RQ5 What is the physical interpretation of the two distinct RQSHJE forms, and how do they differ from the standard QSHJE?

Key findings

• The RQSHJE for spin-1/2 particles is derived as two distinct equations: one for $S_0$ and one for $Z_0$, each corresponding to a different spin projection ($m_s = +1/2$ and $m_s = -1/2$).
• The solution for $S_0$ is expressed as $S_0 = \hbar \arctan(a \theta / \theta_2)$, where $\theta$ is a linear combination of two independent solutions to the $\theta$-equation.
• The RQSHJE for $S_0$ is derived as a second-order differential equation involving the Wronskian and Schwarzian derivative, with terms depending on $E - V + m_0c^2$ and its derivatives.
• The equation reduces to the standard non-relativistic QSHJE in the limit $E \ll m_0c^2$, confirming consistency with known quantum mechanics.
• The RQSHJE for $Z_0$ is structurally analogous, with $E - V + m_0c^2$ replaced by $E - V - m_0c^2$, confirming the dual nature of the spin projection.
• In the classical limit ($\hbar \to 0$), both RQSHJE forms reduce to the classical relativistic Hamilton-Jacobi equation, validating the relativistic and deterministic framework.