dorsal/arxiv
View SchemaFrom a Mechanical Lagrangian to the Schr\"odinger Equation. A Modified Version of the Quantum Newton's Law
| Authors | A. Bouda |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0210193 |
| URL | https://arxiv.org/abs/quant-ph/0210193 |
| DOI | 10.1142/S0217751X03015076 |
| Journal | Int.J.Mod.Phys. A18 (2003) 3347-3368 |
Abstract
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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"abstract": "In the one-dimensional stationary case, we construct a mechanical Lagrangian\ndescribing the quantum motion of a non-relativistic spinless system. This\nLagrangian is written as a difference between a function $T$, which represents\nthe quantum generalization of the kinetic energy and which depends on the\ncoordinate $x$ and the temporal derivatives of $x$ up the third order, and the\nclassical potential $V(x)$. The Hamiltonian is then constructed and the\ncorresponding canonical equations are deduced. The function $T$ is first\nassumed arbitrary. The development of $T$ in a power series together with the\ndimensional analysis allow us to fix univocally the series coefficients by\nrequiring that the well-known quantum stationary Hamilton-Jacobi equation be\nreproduced. As a consequence of this approach, we formulate the law of the\nquantum motion representing a new version of the quantum Newton\u0027s law. We also\nanalytically establish the famous Bohm\u0027s relation % $\\mu \\dot{x} = \\partial S_0\n/\\partial x $ % outside of the framework of the hydrodynamical approach and\nshow that the well-known quantum potential, although it is a part of the\nkinetic term, it plays really a role of an additional potential as assumed by\nBohm.",
"arxiv_id": "quant-ph/0210193",
"authors": [
"A. Bouda"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1142/S0217751X03015076",
"journal_ref": "Int.J.Mod.Phys. A18 (2003) 3347-3368",
"title": "From a Mechanical Lagrangian to the Schr\\\"odinger Equation. A Modified Version of the Quantum Newton\u0027s Law",
"url": "https://arxiv.org/abs/quant-ph/0210193"
},
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