From a Mechanical Lagrangian to the Schr\"odinger Equation. A Modified Version of the Quantum Newton's Law

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a non-relativistic 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 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's law. We also analytically establish the famous Bohm's relation % $\mu \dot{x} = \partial S_0 /\partial x$ % outside of the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, it plays really a role of an additional potential as assumed by Bohm.Comment: 20 pages, LateX, no figure, some calculations are reported in appendice

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

On the Fock Transformation in Nonlinear Relativity

In this paper, we propose a new deformed Poisson brackets which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity. We therefore derive the corresponding momentum transformation which is revealed to be different from previous results. Contrary to the earlier version of Fock's nonlinear relativity for which plane waves cannot be described, our resulting algebra keeps invariant for any coordinate and momentum transformations the four dimensional contraction $p_{\mu} x^{\mu}$, allowing therefore to associate plane waves for free particles. As in Deformed Special Relativity, we also derive a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors. Finally, we establish the dispersion relation for Fock's nonlinear relativity.Comment: 10 pages, no figure