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

A Reformulation of Newton Second Law for Charged Particles and Its Application to Quantum Dynamics

Vu B Ho

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We propose a reformulation of Newton’s second law of motion for charged particles and possible applications of the reformulation to quantum dynamics. We show that the negative energy states arising from the Dirac equation in relativistic quantum mechanics can be verified using the reformulating framework. We also discuss possible hidden dynamics underlying the concept of quantum jumps in quantum mechanics as outlined in Schrödinger’s article: ARE THERE QUANTUM JUMPS? In this case, we show that the hidden dynamics of quantum jumps are also determined by the Coulomb interaction between charged particles.

DOI
10.4236/jamp.2021.911180

Related papers

A Method for Deriving the Dirac Equation from the Relativistic Newton's Second Law

H. Y. Cui

The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton's second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner.

arXiv:quant-ph/0102114•Feb 22, 2001•5 citations

Limitations Of Newton's Second Law In The Quantum Realm: A Comparative Analysis With Quantum Physics

Balakrishna M.T

Newton's second law of motion, F = ma, has long been considered a cornerstone of classical mechanics, providing a fundamental framework for understanding the behavior of macroscopic objects. However, as physics delves into the quantum realm, the applicability of Newton's second law becomes limited, and the principles of quantum mechanics emerge as the dominant description of particle behavior. Beginning with an overview of the historical development of classical mechanics by Newton and the emergence of quantum mechanics in the early 20th century, the article delves into the foundational principles of framework. The classical description of particles as point masses with deterministic trajectories is contrasted with the probabilistic nature of quantum particles described by wave functions and superposition principles. Practical applications and implications of these theories are also discussed, shedding light on their significance in various scientific disciplines. This research article presents a comparative analysis of Newton's second law of motion and quantum physics, examining their fundamental principles and implications through the lens of classical and quantum mechanics. Beginning with an overview of Newton's second law and its significance in classical mechanics, the article delves into the foundational postulates of quantum mechanics and their departure from classical concepts. This explores the inherent limitations of Newton's second law when applied to quantum phenomena, comparing and contrasting its principles with those of quantum physics. Through a comprehensive analysis of evidence and theoretical frameworks, this article elucidates the boundaries of classical mechanics and highlights the need for quantum mechanics to describe phenomena at the atomic and subatomic levels. Through an exploration of key differences and similarities, this aims to provide insights into the behavior of particles at both macroscopic and microscopic scales

Educational Administration Theory and Practice journal•Apr 4, 2024•0 citations

Towards a Relativistic Quantum Mechanics

J. Iliopoulos, Theodore N. Tomaras

Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.

•Oct 25, 2021•0 citations

Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i

Mark C. Palenik

Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations $F=\frac{dP}{dt}$, $P=m\frac{dV}{dt}$, and $\left[X,P\right]=i$. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

arXiv:1402.0396•European Journal of Physics•May 9, 2014•4 citations

Ehrenfest theorem in relativistic quantum theory*

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.

arXiv:2111.10798•Journal of Physics B Atomic Molecular and Optical Physics•Apr 8, 2022•3 citations

Non-Markovian dynamics and quantum jumps

Jyrki Piilo, Sabrina Maniscalco, Kari Härkönen +1 more

Experiments with trapped particles have demonstrated the existence of quantum jumps and the discrete nature of single-system dynamics in quantum mechanics. The concept of jumps is also a powerful tool for simulating and understanding open quantum systems. In non-Markovian systems jump probabilities can become negative due to memory effects between the system and its environment. We discuss a recently presented method that can handle both positive and negative probabilities and provides powerful insight into the dynamics of open systems with memory. The key element is a reversed quantum jump to a system state that was, in principle, already destroyed by an earlier normal jump. Instead of using artificial extensions of the system or exploiting hidden variables we take advantage of the information stored in the quantum ensemble itself.

Optics and Spectroscopy•Mar 1, 2010•1 citation
We propose a reformulation of Newton’s second law of motion for charged particles and possible applications of the reformulation to quantum dynamics. We show that the negative energy states arising from the Dirac equation in relativistic quantum mechanics can be verified using the reformulating framework. We also discuss possible hidden dynamics underlying the concept of quantum jumps in quantum mechanics as outlined in Schrödinger’s article: ARE THERE QUANTUM JUMPS? In this case, we show that the hidden dynamics of quantum jumps are also determined by the Coulomb interaction between charged particles.
The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton's second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner.
Newton's second law of motion, F = ma, has long been considered a cornerstone of classical mechanics, providing a fundamental framework for understanding the behavior of macroscopic objects. However, as physics delves into the quantum realm, the applicability of Newton's second law becomes limited, and the principles of quantum mechanics emerge as the dominant description of particle behavior. Beginning with an overview of the historical development of classical mechanics by Newton and the emergence of quantum mechanics in the early 20th century, the article delves into the foundational principles of framework. The classical description of particles as point masses with deterministic trajectories is contrasted with the probabilistic nature of quantum particles described by wave functions and superposition principles. Practical applications and implications of these theories are also discussed, shedding light on their significance in various scientific disciplines. This research article presents a comparative analysis of Newton's second law of motion and quantum physics, examining their fundamental principles and implications through the lens of classical and quantum mechanics. Beginning with an overview of Newton's second law and its significance in classical mechanics, the article delves into the foundational postulates of quantum mechanics and their departure from classical concepts. This explores the inherent limitations of Newton's second law when applied to quantum phenomena, comparing and contrasting its principles with those of quantum physics. Through a comprehensive analysis of evidence and theoretical frameworks, this article elucidates the boundaries of classical mechanics and highlights the need for quantum mechanics to describe phenomena at the atomic and subatomic levels. Through an exploration of key differences and similarities, this aims to provide insights into the behavior of particles at both macroscopic and microscopic scales
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.
Experiments with trapped particles have demonstrated the existence of quantum jumps and the discrete nature of single-system dynamics in quantum mechanics. The concept of jumps is also a powerful tool for simulating and understanding open quantum systems. In non-Markovian systems jump probabilities can become negative due to memory effects between the system and its environment. We discuss a recently presented method that can handle both positive and negative probabilities and provides powerful insight into the dynamics of open systems with memory. The key element is a reversed quantum jump to a system state that was, in principle, already destroyed by an earlier normal jump. Instead of using artificial extensions of the system or exploiting hidden variables we take advantage of the information stored in the quantum ensemble itself.