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AIP conference proceedings

Brief review on nonlinear master Schrödinger equation and its related gauge invariance

Teguh Budi Prayitno

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We review the derivation as well as the free particle solution of nonlinear master Schrödinger equation. Even though this equation is founded by the Hamilton-Jacobi and continuity equations, the plane-wave solution does still exist for the free particle case. In addition, we will also formulate the gauge invariance form, which describes the electromagnetic interaction. In this case, the Hamilton-Jacobi equation should be modified while the continuity equation remains unchanged.

DOI
10.1063/5.0037849

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We review the derivation as well as the free particle solution of nonlinear master Schrödinger equation. Even though this equation is founded by the Hamilton-Jacobi and continuity equations, the plane-wave solution does still exist for the free particle case. In addition, we will also formulate the gauge invariance form, which describes the electromagnetic interaction. In this case, the Hamilton-Jacobi equation should be modified while the continuity equation remains unchanged.
The solution of free particle has been studied by solving nonlinear master Schrodinger equation. This equation actually consists of two properties, Hamilton-Jacobi equation and continuity equation which describe two classical balance equations. In other words, those two equations were assumed to be hold in quantum level. Unfortunately, there are several properties in ordinary quantum mechanics must be ignored, such as first principle quantization and superposition principle. In this paper we briefly review some important study of the characteristics of nonlinear master Schrodinger equation. In addition, we also have proposed two kinds of free particle solution for the nonlinear master Schrodinger and founded that the solution poses a plane wave solution. Keywords: nonlinear master Schrodinger, free particle, Hamilton-Jacobi.
The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.
We have discussed the quantum particle model for the case of a free particle solution in the one-dimensional Klein-Gordon nonlinear equations. This model was obtained through the two equations from the conservation laws of the classical physics, namely, the Hamilton-Jacobi equation for the relativistic motion and continuity equations. In this case, the Hamilton-Jacobi equation describes a part of the particle while the continuity equation describes the wave side. The derivation of this equation did not use the two postulates in the quantum mechanics, namely, the Einstein and de Broglie’s postulates regarding the quantization of energy and momentum. According to this derivation, the particle side has almost most of the energy of quantum particle that accumulate at a point while the wave part has only a small portion of the energy of the quantum particle that surrounds the part of the particle. In addition, this paper also shows the form of mathematical functions that represent the particle and wave parts for a free particle solution. This form is obtained through a special solution of the free particle which is a plane-wave solution.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.