12 research outputs found
Reply to "Comments on Bouda and Djama's 'Quantum Newton's Law'"
In this reply, we hope to bring clarifications about the reservations
expressed by Floyd in his comments, give further explanations about the choice
of the approach and show that our fundamental result can be reproduced by other
ways. We also establish that Floyd's trajectories manifest some ambiguities
related to the mathematical choice of the couple of solutions of
Schr\"odinger's equation.Comment: 8 pages, LateX, no figure. This letter is a reply to the comments
published by E. R. Floyd in Phys. Lett. A296 (2002) 307-311, quant-ph/020611
The Quantum Newton's Law
Using the quantum Hamilton-Jacobi equation within the framework of the
equivalence postulate, we construct a Lagrangian of a quantum system in one
dimension and derive a third order equation of motion representing a first
integral of the quantum Newton's law. We then integrate this equation in the
free particle case and compare our results to those of Floydian trajectories.
Finally, we propose a quantum version of Jacobi's theorem.Comment: 10 pages, LateX, no figures, minor change
Trajectories in the Context of the Quantum Newton's Law
In this paper, we apply the one dimensional quantum law of motion, that we
recently formulated in the context of the trajectory representation of quantum
mechanics, to the constant potential, the linear potential and the harmonic
oscillator. In the classically allowed regions, we show that to each classical
trajectory there is a family of quantum trajectories which all pass through
some points constituting nodes and belonging to the classical trajectory. We
also discuss the generalization to any potential and give a new definition for
de Broglie's wavelength in such a way as to link it with the length separating
adjacent nodes. In particular, we show how quantum trajectories have as a limit
when the classical ones. In the classically forbidden regions,
the nodal structure of the trajectories is lost and the particle velocity
rapidly diverges.Comment: 17 pages, LateX, 6 eps figures, minor modifications, Title changed,
to appear in Physica Script
The Quantum Reduced Action In Higher Dimensions
The solution with respect to the reduced action of the one-dimensional
stationary quantum Hamilton-Jacobi equation is well known in the literature.
The extension to higher dimensions in the separated variable case was proposed
in contradictory formulations. In this paper we provide new insights into the
construction of the reduced action. In particular, contrary to the classical
mechanics case, we analytically show that the reduced action constructed as a
sum of one variable functions does not contain a complete information about the
quantum motion. In the same context, we also make some observations about
recent results concerning quantum trajectories. Finally, we will examine the
conditions in which microstates appear even in the case where the wave function
is complex.Comment: 12 pages, no figur
The Relativistic Quantum Motions
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.Comment: Latex, 18 pages, 3 eps figure