STM-D-1152Paper1966Published and peer-reviewed

Derivation of the Schrödinger Equation from Newtonian Mechanics

Edward Nelson

Abstract and summary · read the original at the source · APS default licence for the version of record — no Creative Commons licence

In one page

Edward Nelson, a Princeton mathematician, asks what happens if every particle is jostled by a ceaseless random motion — Brownian motion with no friction at all, in empty space or, as he puts it, the ether — while still obeying Newton's law that force equals mass times acceleration. He sets the strength of the jostling so that it is inversely proportional to the mass and fixes its constant by Planck's constant. Then he follows the mathematics. The equations of motion he gets are nonlinear, but a simple change of variables turns them exactly into the Schrödinger equation, and every solution of the Schrödinger equation arises this way. In this picture the hydrogen ground state is a dynamical equilibrium: the electron's random motion and the pull of the nucleus balance, and the usual energy levels come out. Particles keep continuous trajectories. For measurements that reduce to position readings, he shows, the two theories predict the same thing.

Why it matters hereChapter 2's hydrogen balance is Puthoff's route to a stable ground state: a real electromagnetic zero-point field that the electron absorbs from as fast as it radiates. Nelson's paper is the second classical route, and seeing both side by side shows exactly what each one assumes. Both describe the ground state as a balance with a restless background; they differ over what that background is, and that difference is the live research question the chapter points to.

What it claims

01The hypothesis. Every particle of mass m undergoes Brownian motion with a diffusion coefficient inversely proportional to its mass and no friction, the absence of friction being required so that absolute rest cannot be distinguished from uniform motion. The external force acts through Newton's law with the mean acceleration of the particle, as in the Ornstein–Uhlenbeck theory of macroscopic Brownian motion, while the kinematics are those of the Einstein–Smoluchowski theory.Section I, Introduction; Section III, The Hypothesis of Universal Brownian Motion
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02The result. The resulting equations of motion for the osmotic and current velocities are nonlinear, but writing the wave function as the exponential of R plus i S, with R and S built from those velocities, makes it satisfy the Schrödinger equation exactly; conversely every normalised solution of the Schrödinger equation arises from such a Markov process. The same holds for systems of several particles and, assuming the generalised momentum including the vector potential is a gradient, in an external electromagnetic field.Section III, the real time-independent and the time-dependent Schrödinger equation, and the external electromagnetic field
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03The hydrogen atom. In the ground state the electron is in dynamical equilibrium between the random force causing its Brownian motion and the attractive Coulomb force of the nucleus; its trajectory is very irregular, it spends most of its time near the nucleus, and it shows a general tendency to move toward the nucleus whichever direction of time is taken. The hypothesis leads to the correct energy levels for the bound states, interpreted as states of dynamical equilibrium.Section I, Introduction; Section III, after the time-independent Schrödinger equation
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04Comparison with quantum mechanics. For experiments reducible to position measurements, stochastic mechanics and quantum mechanics give the same statistical predictions; the descriptions differ only for combinations such as a sum of observables measured in different experiments, whose operational meaning Nelson questions. He notes that the extra information his description carries, continuous trajectories, is not accessible to experiment in this framework.Section IV, Comparison with Quantum Mechanics
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05The scope, stated by the author. The treatment is non-relativistic, for particles without spin, in external fields, and the class of Hamiltonians handled is limited to second order in the momentum; relativity, spin, the statistics of identical particles and systems of infinitely many degrees of freedom are not treated, so no firm conclusions can be drawn. The work is the Fényes–Weizel theory seen from a different point of view.Section IV, final paragraph; Section V, Discussion
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The argument, in the site's own words

1. A restless background, and Newton's law kept

Nelson starts from a picture any physicist already trusts: a pollen grain in water, kicked about by molecules it cannot see. He asks what happens if every particle, electrons included, lives in a background that kicks it in the same way — with two changes. There is no friction, because friction would single out a state of absolute rest. And the size of the kicks falls as the mass rises, so a bowling ball does not visibly tremble while an electron does. The one constant that sets the size of the kicks turns out to be Planck's constant.

2. The Schrödinger equation falls out

Because the motion is jagged, ordinary velocity does not exist, so Nelson defines two averaged velocities, one looking forward in time and one looking back, and a mean acceleration built from both. Newton's law is applied to that mean acceleration. The two equations that result are nonlinear. Package the two velocities into a single complex function and the nonlinearity disappears: the function obeys the Schrödinger equation. Run the argument backwards and every Schrödinger solution corresponds to one of these random motions.

3. What a ground state is, in this picture

For hydrogen, the ground state is a balance: the random motion pushes the electron outward on average and the nucleus pulls it in. That is the same shape of answer Puthoff gives in 1987 — a ground state as an equilibrium with a restless background — reached by a different road. Puthoff's background is a real electromagnetic zero-point field whose absorbed power balances radiation. Nelson's is a universal Brownian motion whose physical origin he leaves open. Which of those pictures, or which combination of them, reproduces the full three-dimensional atom is exactly what the stochastic-electrodynamics simulations on the Puthoff sheet are testing.

4. What would settle it

Nelson names the limits himself: no spin, no relativity, no identical-particle statistics, no fields with infinitely many degrees of freedom. A stochastic account that carries through those extensions, and a physical identification of the background doing the jostling, are the steps that would turn an equivalence into an explanation.

Citation

Edward Nelson, Department of Mathematics, Princeton University, Derivation of the Schrödinger Equation from Newtonian Mechanics, Physical Review 150, 1079–1085 (28 October 1966). DOI 10.1103/PhysRev.150.1079.

The shelf this sits on

Puthoff, Ground state of hydrogen as a zero-point-fluctuation-determined state (1987) — the electromagnetic route to the same balance: /library/stm-58f597c6df
Cole and Zou, Quantum mechanical ground state of hydrogen obtained from classical electrodynamics (2003): /library/stm-e8f166a3ac
Nieuwenhuizen and Liska, Simulation of the hydrogen ground state in stochastic electrodynamics (2015): /library/stm-9e011ff7d1

The way in

https://doi.org/10.1103/PhysRev.150.1079WHICH COPY WAS READ. The version of record, Physical Review volume 150, number 4, pages 1079 to 1085, 28 October 1966, received 21 April 1966 and revised 21 June 1966, was downloaded on 2026-09-13 from the American Physical Society full-text endpoint at harvest.aps.org/v2/journals/articles/10.1103/PhysRev.150.1079/fulltext, verified as a genuine seven-page PDF, and read in full in text form, sections I to V and the endnotes. LICENCE CHECK. The Crossref deposit for this DOI carries one licence entry, content-version vor, pointing to link.aps.org/licenses/aps-default-license, which is not a Creative Commons licence. WHAT THIS PAGE THEREFORE CARRIES. No text of the paper is reproduced, not even the abstract: the summary, the claims and the walkthrough are the site's own prose, and each claim carries a locator to the section of the printed paper so a reader holding the original can check the site against it. CHAPTERS. Filed to chapter 2, beside the Puthoff 1987 hydrogen balance, as the other classical route to a stable ground state.

How to cite it

Edward Nelson (1966) Derivation of the Schrödinger Equation from Newtonian Mechanics. doi:10.1103/PhysRev.150.1079

Where it sits in the curriculum

What the vacuum is