From a Mechanical Lagrangian to the Schrödinger Equation. A Modified Version of the Quantum Newton's Law

Details From a Mechanical Lagrangian to the Schrödinger Equation. A Modified Version of the Quantum Newton's Law From a Mechanical Lagrangian to the Schrödinger Equation. A Modified Version of the Quantum Newton's Law

2002-10-28

arxiv.org/abs/quant-ph/0210193v2

Int.J.Mod.Phys. A18 (2003) 3347-3368

Physics

Quantum Physics

20 pages, LateX, no figure, some calculations are reported in appendices

Scientific paper

10.1142/S0217751X03015076

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.

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