217 research outputs found
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 , which represents
the quantum generalization of the kinetic energy and which depends on the
coordinate and the temporal derivatives of up the third order, and the
classical potential . The Hamiltonian is then constructed and the
corresponding canonical equations are deduced. The function is first
assumed arbitrary. The development of 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 % % 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
, 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
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