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

arXiv:cond-mat/0306699

Schrodinger Equation for Nanoscience

Janina Marciak-kozlowska B, Mirosław Kozłowski

Published Jun 27, 2003• 3 citations
arXivHTMLPDF

Abstract

The second order (in time) Schrodinger equation is proposed. The additional term (in comparison to Schrodinger equation) describes the interaction of particles with vacuum filled with virtual particle-antiparticle pairs (zitterbewegung). Key words: Schrodinger equation; Nanoscience; Zitterbewegung.
DOI
10.48550/arxiv.cond-mat/0306699

Related papers

Perceptions of the Schrodinger Equation

Spyros Efthimiades

The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.

arXiv:1307.6108•Bulletin of the American Physical Society•Jul 23, 2013•0 citations

The Time Independent Schrodinger Equation Without Using Energy as a Starting Point

Francesco R. Ruggeri

The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.

•Jan 5, 2019•0 citations

Classical Mechanics Theory and Schrödinger's Equation: A Derivation of Relations

J. Q. Li, Yanchun Yu

This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.

Journal of Engineering and Technological Sciences•Sep 23, 2025•1 citation

Particles and Antiparticles According to Covariant Schrödinger Equations

Fritz Bopp

Abstract We introduce manifestly covariant Schrödinger equations. This has the advantage that the energies of particles are always positive. Charges of particles and antiparticles have opposite signs. The physical vacuum equals the formal one. - All that occurs because manifestly covariant Schrödinger equations yield pairs of operators which differ only in some signs of eigenvalues: (i) The number of particles and the charge, (ii) the space-time wave vector and the space time momenta, (iii) the kinematical and the dynamical angular momenta.

Zeitschrift für Naturforschung A•Dec 1, 1975•0 citations

Two-Time Quantum Mechanics

Natalia N. Gorobey, Alexander S. Lukyanenko, Inna Lukyanenko

A two-time quantum theory of a system of two particles with the direct electromagnetic interaction based on a quantum version of the action principle is considered. An analog of Schrodinger equation for the system is obtained.

arXiv:1003.2830•Mar 14, 2010•0 citations

Dynamical Coupling between Particle and Antiparticle in Relativistic Quantum Mechanics: A Multistate Perspective on the Energy–Momentum Relation

Guohua Tao

A molecular formalism based on a decomposed energy space constructed by a modular basis of matter and radiation is proposed for relativistic quantum mechanics. In the proposed formalism, matter radiation interactions are incorporated via the dynamical transformation of the coupled particle/antiparticle pair in a multistate quantum mechanical framework. This picture generalizes relativistic quantum mechanics at minimal cost, unlike quantum field theories, and the relativistic energy–momentum relation is interpreted as energy transformations among different modules through a multistate Schrödinger equation. The application of two-state and four-state systems using a time-dependent Schrödinger equation with pair states as a basis leads to well-defined solutions equivalent to those obtained from the Klein–Gordon equation and the Dirac equation. In addition, the particle–antiparticle relationship is well manifested through a particle conjugation group. This work provides new insights into the underlying molecular mechanism of relativistic dynamics and the rational design of new pathways for energy transformation.

Symmetry•Aug 25, 2023•2 citations
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.
This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.
Abstract We introduce manifestly covariant Schrödinger equations. This has the advantage that the energies of particles are always positive. Charges of particles and antiparticles have opposite signs. The physical vacuum equals the formal one. - All that occurs because manifestly covariant Schrödinger equations yield pairs of operators which differ only in some signs of eigenvalues: (i) The number of particles and the charge, (ii) the space-time wave vector and the space time momenta, (iii) the kinematical and the dynamical angular momenta.
A molecular formalism based on a decomposed energy space constructed by a modular basis of matter and radiation is proposed for relativistic quantum mechanics. In the proposed formalism, matter radiation interactions are incorporated via the dynamical transformation of the coupled particle/antiparticle pair in a multistate quantum mechanical framework. This picture generalizes relativistic quantum mechanics at minimal cost, unlike quantum field theories, and the relativistic energy–momentum relation is interpreted as energy transformations among different modules through a multistate Schrödinger equation. The application of two-state and four-state systems using a time-dependent Schrödinger equation with pair states as a basis leads to well-defined solutions equivalent to those obtained from the Klein–Gordon equation and the Dirac equation. In addition, the particle–antiparticle relationship is well manifested through a particle conjugation group. This work provides new insights into the underlying molecular mechanism of relativistic dynamics and the rational design of new pathways for energy transformation.