Historical research guide · 1152 · 1966
Derivation of the Schrödinger Equation from Newtonian Mechanics
Edward Nelson
In one minute
Nelson examines a model in which every particle undergoes frictionless Brownian motion while its mean acceleration obeys Newton’s law. He sets the diffusion strength inversely proportional to mass and fixes its scale with Planck’s constant. The resulting equations for the current and osmotic velocities are nonlinear, but a change of variables turns them into the Schrödinger equation; every normalized Schrödinger solution has a corresponding Markov process. In this account, the hydrogen ground state is a balance between random motion and the nucleus’s Coulomb attraction, with the usual bound-state energies. Nelson reports the same statistical predictions as quantum mechanics for experiments reducible to position measurements. The paper is nonrelativistic, spinless, and limited to a specified class of Hamiltonians.
Connection
Chapter 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.
From the source
The 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
The 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
The 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
Comparison 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
The 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
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.150.1079How to cite it
Edward Nelson (1966) Derivation of the Schrödinger Equation from Newtonian Mechanics doi:10.1103/PhysRev.150.1079