Relativistic quantum motion

Abstract

Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.

The 3D quantum law of motion and the 3D quantum trajectories

“…For the case of the central potential, the separation of the variables in the reduced action still valid and one can prove it by proceeding by the same way as in Ref. [5]. By replacing Eq.…”

“…(Expressing the total reduced action as a sum of three 1D reduced actions is argued in Ref. [5] in the case of cartesian symmetry potentials. For the case of the central potential, the separation of the variables in the reduced action still valid and one can prove it by proceeding by the same way as in Ref.…”

“…In this figure, the electron moves between two radial extremities r = 0 and r = 2 a o , and describes different trajectories for the same state. All these trajectories pass throw some points constituting nodes with the same manner as for the 1D motions [2,3]. The radial position r = 0 of the nodes correspond to the zero of the radial function X (10) 1…”

“…However, we should plot the 3D QTs when it is possible in order to understand the behavior of the quantum particle's motion in 3D spaces. With this aim, we should proceed exactly as we have done for the 1D quantum and relativistic QTs [2,3]. Indeed, we must first solve the 3D SE to find the functions Ψ 1 and Ψ 2 , which we take into the expression of the reduced action (Eq.…”

Reduced quantum actions for symmetrical and some separated variables potentials cases

“…In the last decade, in the same school of trajectory representation [1,2,3,4,5,7,8,9] a new approach of quantum mechanics has raised from our works [10,11,12] and developed in order to construct a deterministic dynamical approach of quantum mechanics. We started to build a consistent one dimensional theory, then we generalized our approach to 3D systems in earlier works [13,14,15].…”