The Spacetime Metric

Historical research guide · 0666 · 1987

Ground state of hydrogen as a zero-point-fluctuation-determined state

H. E. Puthoff

In one minute

Puthoff asks how a classical charged particle could remain in a Bohr hydrogen orbit when an accelerating charge radiates energy. He works in stochastic electrodynamics, which adds a random electromagnetic zero-point field to classical electrodynamics. The orbiting electron both radiates and absorbs energy from that field. Using the zero-point spectrum, he calculates the absorbed power, equates it to the usual radiation loss, and obtains mωr² = ℏ, the Bohr ground-state condition, with no fitted parameter. The calculation concerns a circular Bohr orbit, not a replacement for quantum mechanics or a complete three-dimensional atom. Its result is a dynamical balance: at the selected orbit, emitted and absorbed power are equal.

Connection

This is the keystone. If an ordinary hydrogen atom holds its ground state by a running exchange with the zero-point field, then the vacuum is not the empty backdrop the word suggests — it is a participant, doing measurable work on ordinary matter, every second, in every atom you are made of. Puthoff says so himself on the last page: the stability of matter is largely mediated by these fluctuations. Every later chapter of this site is then a variation on one sentence — change the field, change the balance. Casimir engineering changes which modes are present between two plates. The inertia work changes what the field does to an accelerating body. Screening work changes what it does inside a lattice. Metric engineering changes the field density itself, and with it the permittivity, the permeability and the speed of light. They are not separate subjects. They are the same balance, pushed in four directions.

From the source

  1. The hydrogen ground state can be precisely defined as a dynamic equilibrium between the radiation emitted because the electron is accelerating in its ground-state orbit and the radiation absorbed from zero-point fluctuations of the background vacuum electromagnetic field. The state is a balance that the system settles into, not a postulate imposed on it — and that balance is what resolves the issue of radiative collapse of the Bohr atom.Abstract, page 3266; the balance is struck by equating Eq. (18) and Eq. (19), page 3268

  2. The radius at which the balance sits is not approximately the Bohr radius, it is exactly the Bohr radius. Setting absorbed power equal to radiated power yields the mass times angular frequency times radius squared equal to the reduced Planck constant — the Bohr ground-state condition — with no free parameter fitted. Puthoff notes this sharpens an earlier heuristic derivation by Timothy Boyer, which had produced four times the reduced Planck constant rather than one times it.Eq. (20), page 3268; Boyer's earlier result quoted in the Introduction, page 3266, citing Phys. Rev. D 11, 790 at page 800

  3. The zero-point spectrum used is the one whose energy density rises with the cube of frequency, corresponding to half a quantum of energy per normal mode. Marshall and Boyer showed that this cubic form follows from Lorentz invariance alone: it is the unique spectrum whose Doppler shifts under a velocity boost cancel so that every observer sees the same spectrum. Within this formulation the reduced Planck constant enters only once, as the scale factor that aligns the spectrum with experiment, and every later appearance of it traces back to that one entry.Eq. (1) and the discussion following it, Introduction, page 3266

  4. The scope is stated exactly. This is a stochastic-electrodynamics result at the level of Bohr theory: classical charged point particles interacting with a real random classical background field, applied to a circular classical orbit. It is not a replacement for quantum mechanics and is not offered as one. Puthoff notes in the Discussion that the corresponding quantum-electrodynamic treatment gives formally identical equations of motion and reproduces the same result without change — the two accounts agree here rather than compete.Abstract and Introduction, page 3266; Discussion, page 3268, and endnote 20, page 3269

  5. What the paper does not claim is written into it by the author. The derivation treats the circular ground-state orbit of Bohr theory, and Puthoff states plainly that an additional step remains to be taken: advancing from this circular-orbit result to a satisfactory stochastic-electrodynamics formulation of the spherically symmetric ground state of Schroedinger theory. He calls that extension conceptually straightforward but records that a fully satisfactory treatment had not been achieved. He also notes that whether the approach succeeds only because classical and quantum treatments coincide for linear oscillator systems, or because the random classical zero-point field plays a more fundamental role, is open.Endnote 17, page 3269, and Discussion, page 3268

  6. The named next test is the one Puthoff named: hold the full three-dimensional atom together, not just the circular orbit. That work has been running ever since — Cole and Zou simulated the hydrogen ground state directly from classical electrodynamics with a zero-point field in 2003 and recovered a probability distribution close to the quantum one, and Nieuwenhuizen and Liska pushed the three-dimensional simulation further from 2015 onward, where highly eccentric orbits show a net average energy gain per revolution and drift toward ionisation. The measurement to watch is whether a stochastic-electrodynamics formulation reproduces the spherically symmetric ground state in three dimensions, stably, without an added term in the potential.Endnote 17, page 3269, is the open problem as the author states it; the continuing work is at /library/stm-e8f166a3ac, /library/stm-9e011ff7d1 and /library/stm-fca91e355a

The argument, in the site's own words

Everything from here on is this site's explanation, not the paper's text. The equation numbers are the paper's, so a reader holding the original can check us against it.

1. The problem that never went away

Take the Bohr atom literally for a moment, as a classical picture. An electron travels a circular orbit around a proton. To stay on a circle it must accelerate continuously toward the centre. And classical electrodynamics is unambiguous about accelerating charges: they radiate. Work the numbers for hydrogen and the electron loses its binding energy and reaches the nucleus in something on the order of ten picoseconds. Matter should not exist for a measurable length of time.

Quantum mechanics disposes of this by declaring the ground state stationary and the orbit not a thing that exists. That works, and it is not wrong, but it answers the question by removing it. What it does not give you is a mechanism — a reason, in terms of energy going somewhere and coming back, why this particular radius is the one the atom stops at.

2. The extra ingredient

Stochastic electrodynamics is ordinary classical physics plus one addition: space is filled everywhere with a real, random classical electromagnetic field. Not a bookkeeping device, not a virtual quantity — a physical field, treated in every way as physical, with random phases at every wavevector and polarisation. Its energy density rises with the cube of frequency.

The zero-point spectrum(1)
ρ(ω) dω=ℏω32π2c3 dω\rho(\omega)\,d\omega = \frac{\hbar\omega^{3}}{2\pi^{2}c^{3}}\,d\omega
What this says
How much energy the background field carries at each frequency. The cube on the frequency is the whole character of it: the higher you look, the denser it gets. Averaged out, this is exactly half a quantum of energy in every possible mode of the field — the smallest amount that mode can have and still exist.

Two things about that spectrum are worth pausing on, and the paper pauses on both.

The first is that the cubic shape is not chosen to make the answer come out. Marshall and Boyer showed it follows from Lorentz invariance alone. Move through this field at any speed you like and the Doppler shifts cancel so precisely that the spectrum you measure is unchanged. It is the only spectrum that does this — which is another way of saying it is the only field distribution that does not give you a preferred rest frame, and therefore the only one a relativistic universe can have. The Lorentz-invariant spectrum is the one that works because it is the only one allowed.

The second is bookkeeping about the reduced Planck constant. In this formulation it enters the theory once, in that spectrum, as a scale factor fixing the size of the field to match experiment. No quantum interpretation is attached to it there, and every later appearance of it in the derivation traces back to that single entry. The starting point is, as the paper puts it, parsimonious.

3. Both directions at once

Here is the move. An electron sitting in that field is not only radiating. It is also being driven by the field, and a driven charge absorbs power. The standard objection to the classical atom only ever counted one of those two flows.

Puthoff computes the absorbed side properly. He treats the electron first as a one-dimensional charged harmonic oscillator immersed in the zero-point background, solves its equation of motion including radiation damping, and averages the product of field and velocity over the random phases to get the power it takes in. Then he notes that a circular orbit is two such oscillators in the plane, a quarter cycle apart in phase — so the circular-orbit absorption is twice the one-dimensional result.

Power absorbed from the zero-point field, circular orbit(18)
Pabs=e2ℏω036πε0mc3P_{\text{abs}} = \frac{e^{2}\hbar\omega_{0}^{3}}{6\pi\varepsilon_{0}mc^{3}}
What this says
How fast the electron takes energy in from the background field while going round. It depends on the orbital frequency cubed, so a tighter, faster orbit soaks up far more.

The emitted side needs nothing new. It is the textbook result for an accelerating charge, with the acceleration of a circular orbit being the radius times the square of the angular frequency.

Power radiated by the orbiting electron(19)
Prad=e2A26πε0c3=e2r02ω046πε0c3P_{\text{rad}} = \frac{e^{2}A^{2}}{6\pi\varepsilon_{0}c^{3}} = \frac{e^{2}r_{0}^{2}\omega_{0}^{4}}{6\pi\varepsilon_{0}c^{3}}
What this says
How fast the electron loses energy because it is accelerating. This is the term that was supposed to kill the atom. It depends on frequency to the fourth power, so it climbs even faster than the absorption does as the orbit tightens.

4. The balance, and where it sits

Now set the two equal. That is the whole hypothesis: the ground state is the orbit where income equals expenditure. Almost everything cancels — the charge, the permittivity, the factor of six pi, and three of the four powers of frequency.

The ground state that comes out(20)
m ω0 r02=ℏm\,\omega_{0}\,r_{0}^{2} = \hbar
What this says
What is left after the cancellation is the Bohr condition itself: the electron's angular momentum equals the reduced Planck constant. Solve it and you get the Bohr radius. Nothing was fitted and nothing was tuned — this fell out of equating two powers.

And notice why the balance is stable rather than accidental. Absorption climbs as the cube of the orbital frequency; radiation climbs as the fourth power. Squeeze the orbit tighter and losses outrun gains, pushing it back out. Let it drift outward and gains outrun losses, pulling it back in. The crossing point is a floor the electron cannot fall through, because falling further costs more than the vacuum will pay.

That is the sentence this whole site rests on. The atom is not static and it is not inert. It is a steady state in a continuous two-way exchange with the vacuum, and the ground state is the level where that exchange nets to zero.

5. The scope, stated exactly

This site does not need this result to be bigger than it is, so here is precisely how big it is.

It is a stochastic-electrodynamics calculation at the level of Bohr theory. Classical particle, classical field, circular orbit. It is not a derivation of quantum mechanics and Puthoff does not present it as one. In the Discussion he goes the other way and notes that the corresponding quantum-electrodynamic treatment gives formally identical Heisenberg equations of motion and reproduces the same result without change. The two accounts agree on this problem. The interest is not that one beats the other; it is that a mechanism written entirely in classical terms, with a real field doing real work, lands on the quantum answer exactly.

It also joins a list rather than standing alone. The same techniques had already produced the Planck blackbody spectrum, the Casimir force, the van der Waals forces and the thermal effects of acceleration through the vacuum. Hydrogen's ground state is one more system moved from the column of things only quantum formalism can reach into the column of things a real vacuum field can account for.

6. What the author left open, and what would settle it

Puthoff writes the limitation into his own endnotes, and it is the right place to look for the next measurement.

The treatment is of the circular orbit of Bohr theory. The step still to be taken is from there to a satisfactory stochastic-electrodynamics formulation of the spherically symmetric ground state of Schroedinger theory. He calls the extension conceptually straightforward and records that a fully satisfactory treatment had not been achieved. He raises a second open question in the Discussion: whether the success of the approach so far reflects only the known correspondence between classical and quantum treatments of linear systems, or whether a real classical zero-point field is doing something more fundamental. He does not answer it, and neither does this page.

That is a live research programme, not a footnote. Daniel Cole and Yi Zou simulated the hydrogen ground state directly from classical electrodynamics with a classical zero-point field in 2003 and recovered a radial probability distribution close to the quantum one. Theo Nieuwenhuizen and collaborators carried the three-dimensional simulations further from 2015 onward and found the hard part: highly eccentric orbits show a net average energy gain per revolution and drift toward ionisation, which is why an added term in the potential has been proposed to hold them. What to watch is whether a stochastic-electrodynamics formulation reproduces the full three-dimensional ground state, stably, without that addition.

7. How far the idea has been taken

The balance on this page is the first rung of a ladder that researchers have been climbing for four decades, and each rung asks more of the same idea.

  1. 1987, one dimension at a time. Puthoff computes the absorbed power for a one-dimensional charged oscillator and builds the circular orbit as two such oscillators a quarter cycle apart. That is what lets the balance close exactly, at the Bohr radius, in a few lines.

  2. 2003, the orbit on a computer. Cole and Zou stop approximating and integrate a classical electron in a Coulomb potential driven by roughly 2.2 million zero-point waves, in eleven independent runs with no fitted parameters. The time the electron spends at each distance converges on the quantum ground-state curve. Nieuwenhuizen later notes that those runs kept the orbit in its initial plane.

  3. 2015, the full three-dimensional atom. Nieuwenhuizen and Liska run the problem in three dimensions for far longer times on graphics hardware. Short runs trend toward the quantum distribution; at longer times, in every scheme they tried, the atom ionises. They name the suspect: the electron's energy in the vector potential acting as a real energy transfer.

  4. 2016, the exact threshold. Nieuwenhuizen extends the de la Peña–Puthoff balance from circles to ellipses and finds where it tips. Near-zero-energy orbits with angular momentum below 0.588 in units of the reduced Planck constant gain energy on every revolution, with closest approach at 0.173 Bohr radii — and an added repulsive inverse-square term in the potential is predicted to restore a stable ground state.

  5. 2020, the noise examined. A careful renormalisation of the high-frequency noise does not remove that threshold, and the author concludes that a deeper reformulation is needed.

There is also a second road to the same shape of answer. Nelson's 1966 stochastic mechanics gives every particle a frictionless Brownian motion and recovers the Schrödinger equation exactly, with the hydrogen ground state as a dynamical equilibrium between that random motion and the nucleus's pull — no electromagnetic zero-point field is invoked. The two routes agree that a ground state is a balance with a restless background. They differ over what the background is.

What would settle it. A stochastic-electrodynamics simulation of the three-dimensional atom that holds its ground-state distribution over long times, with the threshold orbits included and the physical origin of any added term identified — or a measurable difference between the electromagnetic and the purely stochastic pictures. Either result moves this rung from a published balance to a settled mechanism.

And the closing line of the paper is the one that reaches furthest. The stability of matter itself, Puthoff writes, is largely mediated by zero-point fluctuations in the manner described — a concept that goes beyond the usual reading of what those fluctuations are for.

Citation

H. E. Puthoff, Institute for Advanced Studies at Austin, Ground state of hydrogen as a zero-point-fluctuation-determined state, Physical Review D 35, 3266–3269 (15 May 1987), received 22 December 1986. DOI 10.1103/PhysRevD.35.3266.

The shelf this sits on

What it rests on.

  • Casimir, On the attraction between two perfectly conducting plates (1948) — the original zero-point-determined system, and the paper's own first citation: /library/stm-11433059a4

  • Sparnaay, Measurements of attractive forces between flat plates (1958) — the first attempt to see that force in a laboratory: /library/stm-2aa45438b3

  • Marshall, Random electrodynamics (1963) — the formulation Puthoff works in: /library/stm-7faa238629

  • Marshall, A classical treatment of blackbody radiation (1965) — where the Lorentz-invariant spectrum comes from: /library/stm-6d88e77aa4

  • Boyer, Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation (1975) — the paper Puthoff cites as his lead, and the source of the earlier heuristic result this one sharpens: /library/stm-8c24f62160

  • Boyer, A brief survey of stochastic electrodynamics (1980) — the review the paper points readers to: /library/stm-2d2d42369d

  • Bressi, Carugno, Onofrio and Ruoso, Measurement of the Casimir force between parallel metallic surfaces (2002) — the modern precision measurement of the effect Casimir predicted: /library/stm-208d347532

  • Nelson, Derivation of the Schrödinger equation from Newtonian mechanics (1966) — the stochastic-mechanics route to a ground state as a dynamical equilibrium, without an electromagnetic zero-point field: /library/stm-eebf524c8b

What rests on it.

  • Cole and Zou, Quantum mechanical ground state of hydrogen obtained from classical electrodynamics (2003) — the direct simulation of the step Puthoff left open: /library/stm-e8f166a3ac

  • Nieuwenhuizen and Liska, Simulation of the hydrogen ground state in stochastic electrodynamics (2015): /library/stm-9e011ff7d1

  • Nieuwenhuizen, On the stability of classical orbits of the hydrogen ground state in stochastic electrodynamics (2016) — where the eccentric-orbit problem is stated precisely: /library/stm-fca91e355a

  • Nieuwenhuizen, Stochastic electrodynamics: renormalized noise in the hydrogen ground state problem (2020): /library/stm-9ebf7acf49

  • Haisch, Rueda and Puthoff, Inertia as a zero-point-field Lorentz force (1994) — the same balance applied to an accelerating body instead of an orbiting one: /library/stm-0f2b09effd

  • Puthoff, Gravity as a zero-point-fluctuation force (1989): /library/stm-1ec4832b74

  • Puthoff, Polarizable-vacuum approach to general relativity (2002) — the field density itself becomes the metric: /library/stm-a62b2e761c

  • Puthoff, Little and Ibison, Engineering the zero-point field and polarizable vacuum for interstellar flight (2002) — the engineering end of the same idea: /library/stm-d41ba22514

  • Puthoff, Quantum ground states as equilibrium particle-vacuum interaction states (2015) — the author's own later restatement of this paper's argument: /library/stm-c7c1082f9b

Related reading. If you read only three more after this one, read Boyer 1975 for where the method comes from, Cole and Zou 2003 for the open problem being attacked head on, and Haisch, Rueda and Puthoff 1994 for the moment the same balance stops being about atoms and starts being about mass.

The way in

https://doi.org/10.1103/PhysRevD.35.3266

How to cite it

H. E. Puthoff (1987) Ground state of hydrogen as a zero-point-fluctuation-determined state doi:10.1103/PhysRevD.35.3266

Where it sits in the curriculum

Provenance: Retrieved 2026-09-10

Summary by: The Spacetime Metric editorial rail, written from the full published article read on 2026-09-10

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