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

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