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Physica Scripta

The 3D quantum stationary Hamilton–Jacobi equation for symmetrical potentials

T. Djama

Published Dec 5, 2006• 3 citations
Publisher

Abstract

We establish the quantum stationary Hamilton–Jacobi equation in 3D and its solutions for three symmetrical potentials: Cartesian symmetry potential, spherical symmetry potential and cylindrical symmetry potential.
DOI
10.1088/0031-8949/75/1/012

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We present the analytic solution for the stationary quantum HamiltonJacobi equation. Knowing the strong relation between the Riccati and quantum Hamilton-Jacobi equations, we develop a simple method to obtain the exact solution. Then, in order to prove the validity of the proposed method, we use two central potentials: the three-dimensional harmonic oscillator and Coulomb potential, both with bound-states. Finally, we compute the action-angle variables in a entirely quantum version for to achieve connect with the nodes of the wave function.
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 QSHJES12QSHJES_{1\over2}QSHJES21​​ have two explicit forms, which represent the two possible projection of the Spin 1/2.
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