Quantum Newton's law

Abstract

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.

Reply to “Comments on Bouda and Djama's ‘Quantum Newton's Law’”

“…That's precisely what we have done in Ref. [2]. Furthermore, if we would take up the Lagrangian depending on (x,ẋ,ẍ,ẋ) and keep the definition…”

“…Firstly, we indicate that the above relation, up to an additive constant, is already written in our paper [2], Eq. (39).…”

Relativistic quantum motion

“…As it is explained in Ref. [10], this is the reason why in problems for whichh can be disregarded, relativistic quantum trajectories reduce to the purely relativistic one [9,10]. It is useful to note that at the classical limit (h → 0), not only the nodes becomes infinitely near, but in addition the paths of all RQTs go to the path of the purely relativistic one (a = 1, b = 0).…”

“…It is a generalization of the one exposed in Refs. [9,10,11]. So, we have derived the fundamental relativistic quantum Newton's law expressed into Eqs.…”

From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law

“…[14]. When the quantum coordinate [4,13] is used to apply Jacobi's theorem [6] or to express the Lagrangian in order to obtain the equation of motion [11], the formalism does not seem to suffer from any mathematical ambiguity. Furthermore, the fundamental result is reproduced with many formulations [11].…”