Quantum Behavior of a Classical Particle Subject to a Random Force

• Published in: Foundations of physics Vol. 51; no. 1
• Main Author: Gokler, Can
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2021; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Classical Mechanics; Harmonic oscillators; History and Philosophical Foundations of Physics; Philosophy of Science; Physics; Physics and Astronomy; Quantum mechanics; Quantum Physics; Relativity Theory; Schrodinger equation; Statistical Physics and Dynamical Systems; Brownian motion; Method of stochastic averaging; Quantum mechanics; Stochastic mechanics
• ISSN: 0015-9018; 1572-9516
• DOI: 10.1007/s10701-021-00422-3

We give a partial answer to the question whether the Schrödinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a random force can be described by the Schrödinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson’s stochastic formulation of quantum mechanics for a random force depending on a small parameter. We show that the same result applies to small potential perturbations around the harmonic oscillator. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables. The main motivation of this paper is to provide a step for constructing a Newtonian theory which would approximately reproduce quantum mechanics both in unitary evolution and measurement regimes.