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SOLUTION OF HARMONIC OSCILLATOR OF NONLINEAR MASTER SCHRÖDINGER

Teguh Budi Prayitno

Published Feb 1, 2012• 0 citations
OpenAlex

Abstract

We have computed the solution of a nonrelativistic particle motion in a harmonic oscillator potential of the nonlinear master Schrödinger equation. The equation itself is based on two classical conservation laws, the Hamilton-Jacobi and the continuity equations. Those two equations give each contribution for the definition of quantum particle. We also prove that the solution can’t be normalized. Keywords : harmonic oscillator, nonlinear Schrödinger.

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We have computed the solution of a nonrelativistic particle motion in a harmonic oscillator potential of the nonlinear master Schrödinger equation. The equation itself is based on two classical conservation laws, the Hamilton-Jacobi and the continuity equations. Those two equations give each contribution for the definition of quantum particle. We also prove that the solution can’t be normalized. Keywords : harmonic oscillator, nonlinear Schrödinger.
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.
In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.
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.