Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

Quantum particle model of free particle solution in one-dimensional Klein-Gordon equation

Teguh Budi Prayitno

Published Jan 1, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.5132667

Related papers

The Free Particle Solution for NonLinear Master Schrödinger

Teguh Budi Prayitno

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.

•Dec 27, 2010•0 citations

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

Teguh Budi Prayitno

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.

AIP conference proceedings•Jan 1, 2021•0 citations

Obtaining the Klein-Gordon wave equation without using the quantum operators

Roshan Raj

A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.

•Oct 30, 2023•0 citations

Hamilton-Jacobi and Lagrange formulation of relativistic quantum mechanics wave equations with solutions with only-positive and only-negative kinetic energies

Luis Grave de Peralta, Arquímedes Ruiz-Columbié

Abstract Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.

•Feb 18, 2022•0 citations

Equivalence of classical Klein-Gordon field theory to correspondence-principle first quantization of the spinless relativistic free particle

Steven Kenneth Kauffmann

It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.

arXiv:1012.5120•Dec 14, 2010•1 citation

Hamilton-Jacobi and Lagrange Formulations of Relativistic Quantum Mechanics Wave Equations with Solutions with Only-Positive and Only-Negative Kinetic Energies

Luis Grave de Peralta, Arquímedes Ruiz-Columbié

Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.

Journal of Modern Physics•Jan 1, 2022•0 citations
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.
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.
A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.
Abstract Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.