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

Journal of Applied Mathematics and Physics

How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

Frank Blume

Published Jan 1, 2023• 0 citations
Publisher

Abstract

We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate-by way of superposition-the totality of solutions of various well-known classical and quantum-mechanical wave equations.

DOI
10.4236/jamp.2023.119178

Related papers

A Derivation of the Path-Integral Formalism of Quantum Mechanics from Probability Conservation along the Velocity Flows of Classical Particles

Frank Blume

Abstract: Denoting by v the velocity flow of a classical particle that is subject to a potential V, we demonstrate that the path-integral formalism of non-relativistic quantum mechanics can be obtained by superimposing wave functions that are solutions of a wave equation which in turn directly corresponds to the probability conservation equation div(ψ ∗ ψv)=0.

Quantum Physics Letters•Jul 29, 2013•1 citation

Relativistic Schrödinger equation and probability currents for free particles

David Carfì

In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.

Proceedings of the International Geometry Center•Jan 13, 2024•2 citations

Remarks on the mathematical origin of wave mechanics and consequences for a quantum mechanics in a gravitational field

Mayeul Arminjon

According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.

arXiv:gr-qc/0203104•Mar 28, 2002•7 citations

Inference of Schrödinger's Equation from Classical-Mechanical Solution

J. X. Zheng-Johansson, Johansson, P-I.

We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field $V$ in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.

arXiv:physics/0411134•Nov 15, 2004•2 citations

Unitary Theory of Quantum Mechanics and General Relativity

Eliade N. Stefanescu

Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.

European journal of applied sciences•Aug 31, 2021•7 citations

A Relativistic Quantum Wave Equation That Reduces to the Schrödinger Equation in the Non-Relativistic Limit

Brian B. K. Min

A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.

•Apr 1, 2015•0 citations
We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate-by way of superposition-the totality of solutions of various well-known classical and quantum-mechanical wave equations.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.
We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field VVV in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.