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

The paper, in the author's own words

The abstract below is reproduced verbatim from the published article. It is the author's own statement of the result, and it is the only text from the paper reproduced on this page.

We show here that, within the stochastic electrodynamic formulation and at the level of Bohr theory, the ground state of the hydrogen atom can be precisely defined as resulting from a dynamic equilibrium between radiation emitted due to acceleration of the electron in its ground-state orbit and radiation absorbed from zero-point fluctuations of the background vacuum electromagnetic field, thereby resolving the issue of radiative collapse of the Bohr atom.

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 (1987), abstract, page 3266. Copyright 1987 The American Physical Society; published under the APS default licence, which is not a Creative Commons licence. No further text from the paper is reproduced here — follow the DOI to read the original.

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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)

$$\rho(\omega)\,d\omega = \frac{\hbar\omega^{3}}{2\pi^{2}c^{3}}\,d\omega$$

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)

$$P_{\text{abs}} = \frac{e^{2}\hbar\omega_{0}^{3}}{6\pi\varepsilon_{0}mc^{3}}$$

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)

$$P_{\text{rad}} = \frac{e^{2}A^{2}}{6\pi\varepsilon_{0}c^{3}} = \frac{e^{2}r_{0}^{2}\omega_{0}^{4}}{6\pi\varepsilon_{0}c^{3}}$$

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)

$$m\,\omega_{0}\,r_{0}^{2} = \hbar$$

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.

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.

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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

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.