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arXiv:0801.3311

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

Alex Granik

Published Jan 22, 2008• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.0801.3311

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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.
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.
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.
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.
The Schr?dinger differential equation is what we usually solve for the microscopic particles in non-relativistic quantum mechanics. Niels Bohr suggested the power two of the (usually) complex answer shows the probability of the particle’s existence at a point of space. Also, the time dependence of Schrodinger wave equation is one whereas for light in electromagnetism is two. In this paper, we show a solution for both problems. We derive a Wave Equation for the energy of every system. This electromagnetic wave equation is shown to convert to those classical (i.e. the Schrodinger) and special relativistic (i.e. Klein-Gordon) quantum mechanical equations. Also, accordingly there definitely is a physical meaning to answer to this wave equation. And therefore, switching the probabilistic interpretation of quantum mechanics to a deterministic one as (Albert) Einstein demanded.