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The Relativistic Quantum Motions.

Published Jan 1, 2004• 0 citations
OpenAlex

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. PACS: 03.65.Bz; 03.65.Ca; Key words: relativistic quantum law of motion, Lagrangian, relativistic quantum stationary Hamilton-Jacobi equation, relativistic velocities.

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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. PACS: 03.65.Bz; 03.65.Ca; Key words: relativistic quantum law of motion, Lagrangian, relativistic quantum stationary Hamilton-Jacobi equation, relativistic velocities.
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.
In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.
Abstract The first calculation, similar to that of Lewis of 1908, takes as postulates Newton’s second law of motion and work-energy equivalence, the second makes use of the Hamilton–Jacobi equation for motion in free space and the third identifies the velocity of a freely moving particle with the group velocity of its associated de Broglie wave. In all cases the equivalence of mass and energy is also postulated. How the path integral formulation of quantum mechanics relates all the different calculations is demonstated. The essential roles of relativity, quantum mechanics and the particulate nature of light in the establishment of the current standard units of length, time and mass, are discussed, as well as the similar relations of the fundamental constants c , h and k to energy at the microscopic physics level.