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The Schr{\"o}dinger Equation: Derivation from Classical Mechanics Resulting in the Wave-Particle Duality and Other Principal Corollaries

Alex Granik

Published Jan 22, 2008• 0 citations
OpenAlex

Abstract

According to the widely accepted notion, the Schr{o}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge of SE and arrive at it by introducing a new postulate - neither present in classical mechanics nor following from experiments - with a suitable but physically unjustifiable choice of a key arbitrary constant.

Related papers

Straightforward Derivation of the Schr{ö}dinger Equation from Classical Mechanics and the Planck Postulate

Alex Granik

According to the widely accepted notion, the Schr{ö}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge of SE and arrive at it by introducing a new postulate - neither present in classical mechanics nor following from experiments - with a suitable but physically unjustifiable choice of a key arbitrary constant.

arXiv:0801.3311•Jan 22, 2008•1 citation

Derivation of Schrodinger's equation from the Hamilton-Jacobi and the Eikonal equations

Lachezar S. Simeonov

Most authors of textbooks on quantum mechanics either postulate or sketch a short `ad hoc` derivation of Schrodinger's equation. In this work we give a detailed derivation of Schrodinger's equation from the Hamilton-Jacobi equation and the Eikonal equation in geometrical optics. We start from the historical debates on the nature of light -- whether it is a beam of particles, or waves in the aether. We derive the Eikonal equation and show the conditions when a wave can behave as a beam of particles. Then we discuss several experiments with an electron gun, that show clearly diffraction and interference of a single electron. Next, in order to explain these experiments, we derive Schrodinger's equation by comparing Hamilton-Jacobi equation in classical mechanics with the Eikonal equation in geometrical optics. To do that, we first show how to derive the wave equation from the Eikonal equation (not the other way around!). Second, we use this method to derive Schrodinger's equation from the Hamilton-Jacobi equation. Next, we derive Born's statistical rule using the early understanding of de Broglie that both particles and waves exist. Afterwards, we show that historically people preferred to remove the particles (as well as their trajectories) altogether from de Broglie's ideas but retained Born's rule (the so called Copenhagen interpretation). These derivations of the foundations of quantum mechanics do not follow precisely the history of the subject. Rather we select some early ideas and experiments in a judicious manner to present Schrodinger's equation in a logical and ordered way. We use the electron gun experiments instead of black body radiation and photoelectric effect. Our derivation may bring more light and satisfaction for the undergraduate students about the confusing and rather mysterious subject of quantum mechanics.

arXiv:2410.03689•Sep 21, 2024•0 citations

Derivation of the Schrödinger equation from the Hamilton–Jacobi equation in Feynman's path integral formulation of quantum mechanics

J.H. Field

It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.

arXiv:1204.0653•European Journal of Physics•Nov 17, 2010•29 citations

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

The time dependent Schrodinger equation revisited I: quantum field and classical Hamilton-Jacobi routes to Schrodinger's wave equation

Marlan O. Scully

The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.

Journal of Physics Conference Series•Feb 1, 2008•11 citations

All roads lead to Schrödinger: projections of the same equation onto different theoretical frameworks of quantum mechanics

J. Oliveira-Cony

Abstract Schrödinger’s equation (SE) stands as one of the most recognized formulas in physics, yet its derivation and interpretation vary widely across different theoretical frameworks of quantum mechanics. In this paper, we explore five distinct approaches to obtaining SE: via the optico-mechanical analogy, a group-theoretic method, the action principle (through Hamilton–Jacobi theory and path integrals), stochastic mechanics, and direct postulation. Each derivation is analysed in light of the conceptual foundations it implies, with special attention to how it aligns with the orthodox, realistic, and statistical interpretations of quantum theory. By comparing these frameworks, it is emphasized the interpretative flexibility of SE and the philosophical implications of treating it as a derived versus a postulated equation. This comparison sheds light on the multiplicity of paths that converge on the same mathematical structure, on the particularities of these structures, and on the interpretive tensions that underlie quantum mechanics as a physical theory.

European Journal of Physics•Dec 2, 2025•0 citations
According to the widely accepted notion, the Schr{o}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge of SE and arrive at it by introducing a new postulate - neither present in classical mechanics nor following from experiments - with a suitable but physically unjustifiable choice of a key arbitrary constant.
According to the widely accepted notion, the Schr{ö}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge of SE and arrive at it by introducing a new postulate - neither present in classical mechanics nor following from experiments - with a suitable but physically unjustifiable choice of a key arbitrary constant.
Most authors of textbooks on quantum mechanics either postulate or sketch a short `ad hoc` derivation of Schrodinger's equation. In this work we give a detailed derivation of Schrodinger's equation from the Hamilton-Jacobi equation and the Eikonal equation in geometrical optics. We start from the historical debates on the nature of light -- whether it is a beam of particles, or waves in the aether. We derive the Eikonal equation and show the conditions when a wave can behave as a beam of particles. Then we discuss several experiments with an electron gun, that show clearly diffraction and interference of a single electron. Next, in order to explain these experiments, we derive Schrodinger's equation by comparing Hamilton-Jacobi equation in classical mechanics with the Eikonal equation in geometrical optics. To do that, we first show how to derive the wave equation from the Eikonal equation (not the other way around!). Second, we use this method to derive Schrodinger's equation from the Hamilton-Jacobi equation. Next, we derive Born's statistical rule using the early understanding of de Broglie that both particles and waves exist. Afterwards, we show that historically people preferred to remove the particles (as well as their trajectories) altogether from de Broglie's ideas but retained Born's rule (the so called Copenhagen interpretation). These derivations of the foundations of quantum mechanics do not follow precisely the history of the subject. Rather we select some early ideas and experiments in a judicious manner to present Schrodinger's equation in a logical and ordered way. We use the electron gun experiments instead of black body radiation and photoelectric effect. Our derivation may bring more light and satisfaction for the undergraduate students about the confusing and rather mysterious subject of quantum mechanics.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
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 .
The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.
Abstract Schrödinger’s equation (SE) stands as one of the most recognized formulas in physics, yet its derivation and interpretation vary widely across different theoretical frameworks of quantum mechanics. In this paper, we explore five distinct approaches to obtaining SE: via the optico-mechanical analogy, a group-theoretic method, the action principle (through Hamilton–Jacobi theory and path integrals), stochastic mechanics, and direct postulation. Each derivation is analysed in light of the conceptual foundations it implies, with special attention to how it aligns with the orthodox, realistic, and statistical interpretations of quantum theory. By comparing these frameworks, it is emphasized the interpretative flexibility of SE and the philosophical implications of treating it as a derived versus a postulated equation. This comparison sheds light on the multiplicity of paths that converge on the same mathematical structure, on the particularities of these structures, and on the interpretive tensions that underlie quantum mechanics as a physical theory.