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

CRC Press eBooks

Schrödinger's Equation

F.J. Duarte

Published Oct 29, 2013• 0 citations
Publisher

Abstract

As its name suggests, Schrodinger’s equation is the brain child of Schrodinger and was first disclosed in 1926 in a paper entitled An Undulatory Theory of the Mechanics of Atoms and Molecules . Schrodinger begins by considering the dynamics of a particle moving through a force field. The Schrodinger’s equation is arrived at in an alternative heuristic path modeled after Haken. The Schrodinger’s equation is a wave equation that incorporates classical particle concepts, and classical wave function concepts, in its derivation. In other words, a heuristic approach to Schrodinger’s equation utilizes Max Planck’s quantum energy, classical kinetic energy of a free particle, and the classical “wave functions of ordinary wave optics.” Schrodinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity.

DOI
10.1201/9781003649434-21

Related papers

The wave function

Donald D. Fitts

Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.

Cambridge University Press eBooks•Aug 26, 1999•0 citations

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

Derivation of the Schrodinger Equation from Classical Physics

José Garrigues Baixauli

In this work, the Schrodinger equation is deduced in a very simple manner. The starting point is the assumption that the Universe and particles are formed by four-dimensional Planck atoms. The wave function is the ratio between the kinetic energy that the electron has when it is unobserved and the energy that it acquires due the observation.

International journal of scientific research•May 1, 2016•2 citations

Derivation of the Schrodinger equation from fundamental principles

Wenzhuo Zhang, Anatoly Svidzinsky

Schrodinger path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.

quant-pharXiv:2603.27041•Mar 31, 2026

Deriving the Schrödinger Equation

Steven M. Pascal

The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .

CRC Press eBooks•Jul 12, 2023•0 citations

Quantum mechanical motion of classical particles

V. D. Rusov, Konstantin Alexandrovich Lukin

The Schrödinger equation for the particle wave function introduced via its action was derived from Newton equation for a point-like particle moving under effect of both potential force and fluctuation-dissipative environment, provided considering only stable motion of the particle. The model considered assumes existence of wave function as a physical field rather than just a mathematical abstraction.

Journal of Physics Conference Series•May 10, 2012•0 citations
As its name suggests, Schrodinger’s equation is the brain child of Schrodinger and was first disclosed in 1926 in a paper entitled An Undulatory Theory of the Mechanics of Atoms and Molecules . Schrodinger begins by considering the dynamics of a particle moving through a force field. The Schrodinger’s equation is arrived at in an alternative heuristic path modeled after Haken. The Schrodinger’s equation is a wave equation that incorporates classical particle concepts, and classical wave function concepts, in its derivation. In other words, a heuristic approach to Schrodinger’s equation utilizes Max Planck’s quantum energy, classical kinetic energy of a free particle, and the classical “wave functions of ordinary wave optics.” Schrodinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
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 Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .