A new look at the derivation of the Schrödinger equation from Newtonian mechanics

• Published in: Annalen der Physik Vol. 12; no. 6; pp. 371 - 402
• Main Authors: Fritsche, L., Haugk, M.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 23.07.2003; WILEY‐VCH Verlag
• Subjects: Schrödinger equation; stochastic quantum mechanics; vacuum fluctuations
• ISSN: 0003-3804; 1521-3889
• DOI: 10.1002/andp.200310017

We present a modified version of Nelson's seminal paper on the derivation of the time‐dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation $\Delta E \Delta t\approx \hbar/2$ where Δ t is the average time for Δ E to persist. The stability of atoms, the zero‐point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.