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Schrödinger equation derived from stochastic electrodynamics

L. de la Peña-Auerbach, Ana Marı́a Cetto

Physics Letters ASep 1, 19775 citations

Stochastic mechanics and nonlinear schrödinger equation

Hans-Jürgen Mann

Reports on Mathematical PhysicsAug 1, 19994 citations

Derivation of the Schrödinger equation IV: Stochastic and the Langevin Equations for Quantum Mechanics

Olavo Leopoldino da Silva Filho, Marcello Caio Ramon e Barros Ferreira

This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.

Revista Brasileira de Ensino de FísicaJan 1, 20241 citation

Stochastic Mechanics & Classical Mechanics with Finite Differences

Robert F. Bordley

Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.

Journal of the Physical Society of JapanOct 15, 19970 citations

Derivation and application of quantum Hamilton equations of motion

Jeanette Köppe, Wilfried Grecksch, Wolfgang Paul

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. imageNon‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image

Annalen der PhysikDec 13, 201614 citations

The Stochastic Schrödinger Equation

Crispin W. Gardiner, P. Zoller

Springer series in synergeticsJan 1, 20001 citation

Derivation of the Schrödinger Equation from Classical Stochastic Dynamics

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