Quantum ground states as equilibrium particle–vacuum interaction states

Abstract

A remarkable feature of atomic ground states is that they are observed to be radiationless in nature, despite (from a classical viewpoint) typically involving charged particles in accelerated motions. The simple hydrogen atom is a case in point. This universal ground-state characteristic is shown to derive from particle–vacuum interactions in which a dynamic equilibrium is established between radiation emission due to particle acceleration, and compensatory absorption from the zero-point fluctuations of the vacuum electromagnetic field. The result is a net radiationless ground state. This principle constitutes an overarching constraint that delineates an important feature of quantum ground states.

Keywords. Quantum ground states · Vacuum fluctuations · Particle–vacuum interaction states · Zero-point fluctuations · Harmonic oscillator quantum ground state

1. Introduction

One of the apparent paradoxes of quantum theory that students often query is the radiationless nature of atomic ground states. The paradox lies in the fact that, classically, radiation that might be anticipated to occur from accelerated charged-particle motions in atomic ground states is not observed. In the hydrogen atom, for example, the orbiting electron does not radiate its energy away and spiral into the nucleus. The fact that during decades of successful application of quantum theory we have come to take for granted the radiationless feature of these special "bottom-rung" stationary states does not in any way detract from this remarkable property. Fortunately, a rapprochement between classical and quantum viewpoints with regard to ground states is possible.

When addressed in analytical detail it becomes clear wherein the resolution to this apparent paradox lies. It is that one must properly take into account how charged-particle ground-state motions interact with the vacuum, specifically the zero-point fluctuations of the vacuum electromagnetic field. Although such considerations are not usually invoked in the day-to-day application of quantum theory to ground-state specification, the argument that follows clarifies that the standard quantum formalism with its associated radiationless ground states has its genesis in the dynamics of underlying particle–vacuum interactions and that the vacuum field is in fact formally necessary for the stability of atoms in quantum theory. As summarized in an earlier paper addressing spontaneous emission processes, by Fain: "The crucial role of the vacuum fluctuations emerges in the ground state of matter. The stability of the ground state (i.e., the fact that it does not radiate) is purely a quantum effect which is due to the vacuum fluctuations."

Setting aside a full quantum mechanical treatment, it is sufficient for heuristic purposes to treat such problems semiclassically on the basis of point particles interacting with a random, classical radiation field whose spectral characteristics are those of the known quantum vacuum zero-point fluctuation distribution, an approach known as stochastic electrodynamics. Early detailed pedagogical papers by Boyer provide a broad foundation for the stochastic electrodynamics approach as applicable to many problems generally considered to require quantum mechanical treatment. The further development of the approach in the decades following is well summarized by de la Peña and Cetto. The treatment in this paper extends earlier work by the author on the quantum ground state of hydrogen to address the case for quantum ground states in general. The stochastic electrodynamics ansatz takes advantage of the fact that derivations of vacuum-fluctuation-driven phenomena based on multipole and radiation-field interactions parallel Heisenberg-picture derivations in standard quantum electrodynamics. Specifically, despite shortcomings in the approach as a proposed alternative to quantum theory, in the cases considered here the calculations are analogous to quantum electrodynamical calculations with a symmetric ordering of photon creation and annihilation operators. Before considering application on the basis of a general formalism, let us apply the central argument in detail to the simple one-dimensional harmonic oscillator.

2. Nonrelativistic harmonic oscillator

For a one-dimensional harmonic oscillator of natural frequency omega-zero located at the origin and immersed in the vacuum zero-point field, the nonrelativistic Abraham–Lorentz equation of motion for a point particle of mass m and charge q, including radiation damping, is Equation 1 in the source: the mass times the second time derivative of the displacement, plus the mass times the square of the natural frequency times the displacement, equals the charge squared divided by six pi times the permittivity of free space times the cube of the speed of light, all multiplying the third time derivative of the displacement, plus the charge times the component of the vacuum zero-point electric field along the direction of motion. Here the force contribution from the magnetic field is neglected; it is restored in Section 3.

The required expression for the electric field is obtained as a component of the isotropic, homogeneous electromagnetic vacuum zero-point distribution whose spectral energy density is given by the Lorentz-invariant expression, Equation 2 in the source: the energy density per unit frequency interval equals Planck's reduced constant times the cube of the frequency, divided by two pi squared times the cube of the speed of light. This corresponds to an energy of half Planck's reduced constant times the frequency per normal mode.

In the stochastic electrodynamics ansatz the Fourier composition underlying this spectrum can be written as a sum of plane waves, Equation 3 in the source: the zero-point electric field is the real part of a sum over the two orthogonal polarizations of an integral over the three-dimensional wavevector, of the polarization unit vector multiplied by the square root of Planck's reduced constant times the frequency divided by eight pi cubed times the permittivity of free space, multiplied by the exponential of the imaginary unit times the wavevector dotted into position, minus the imaginary unit times the frequency times time, plus the imaginary unit times a random phase depending on the wavevector and the polarization. A similar expression for the magnetic field is obtained by replacing the electric field with the magnetic field, the polarization unit vector with the cross product of the unit wavevector and the polarization vector, and the permittivity of free space with the permeability of free space. In these expressions the two polarizations are orthogonal, the polarization and propagation unit vectors are orthogonal to one another, the random phases are distributed uniformly in the interval zero to two pi and independently for each wavevector and polarization, and the frequency is the wavenumber times the speed of light.

Substitution of Equation 3 into Equation 1 leads to expressions for displacement, velocity and acceleration — Equations 4, 5 and 6 in the source. Each has the same structure: the charge over the mass, multiplying the real part of the same sum over polarizations and integral over wavevector, with the projection of the polarization vector on the direction of motion, the same square-root amplitude factor, and a denominator D; for the displacement the factor is one over D, for the velocity it is minus the imaginary unit times the frequency over D, and for the acceleration it is minus the frequency squared over D. Equation 7 in the source defines the denominator D as minus the frequency squared, plus the square of the natural frequency, minus the imaginary unit times the cube of the frequency times the damping parameter. Equation 8 in the source defines the damping parameter as the charge squared divided by six pi times the permittivity of free space, the mass and the cube of the speed of light.

Now, for bounded, steady-state motion, we assume stationary expectation values for the mean-square position variable and its time derivatives. Equation 9 in the source writes the mean square position as a double sum over polarizations and a double integral over two wavevectors, of the product of the two projection factors, the two amplitude factors, and one over the product of D with the complex conjugate of the second denominator, multiplying one half of the real part of the exponential of the imaginary unit times the difference of the wavevectors dotted into position, minus the imaginary unit times the difference of the frequencies times time, plus the difference of the two random phases; the complex conjugate and the factor of one half derive from the use of exponential notation.

Writing the volume element in wavevector space as the solid angle element times the differential of the wavenumber times the wavenumber squared, and averaging over random phases, Equation 10 in the source states that the average of that exponential is the product of a Kronecker delta in the polarizations, a Kronecker delta in the frequencies, and a three-dimensional Dirac delta function in the difference of the wavevectors. Equation 9 can therefore be simplified to Equation 11 in the source: one half of the charge squared over the mass squared, times an integral over solid angle and wavenumber of the wavenumber squared, the amplitude factor squared divided by the product of D with its complex conjugate, and the sum over polarizations of the squared projection factor.

We further note that the sum over polarizations of the products of the projections of the two polarization vectors on any two directions is, by Equation 12 in the source, the Kronecker delta of those directions minus the product of the projections of the unit wavevector on them. Consequently the angular integration in wavevector space, Equation 13 in the source, gives the integral over solid angle of one minus the squared projection of the unit wavevector, which equals eight pi over three.

Substitution of Equation 13 into Equation 11, and a change of variables from wavenumber to frequency, then leads to Equation 14 in the source: the mean square position equals the charge squared times Planck's reduced constant, divided by six pi squared times the permittivity of free space, the mass squared and the cube of the speed of light, multiplying the integral from zero to infinity of the cube of the frequency divided by the product of D with its complex conjugate — that product being the square of the difference between the squared natural frequency and the squared frequency, plus the damping parameter squared times the sixth power of the frequency.

Due to the smallness of the damping parameter for charge-to-mass ratios of interest, the integrand in Equation 14 is sharply peaked around the natural frequency. We can therefore invoke the standard resonance approximation, extending the limits of integration and replacing the frequency by the natural frequency in all but the difference term. This yields, with substitution of the definition of the damping parameter from Equation 8, the mean square fluctuation in position as Equation 15 in the source: an integral of Lorentzian lineshape form, whose value is unity, multiplied by Planck's reduced constant divided by twice the mass and the natural frequency. So the mean square position is Planck's reduced constant over twice the mass times the natural frequency.

Calculation of the mean square fluctuation in velocity follows as above, except that one over the product of D with its complex conjugate is replaced by the frequency squared over that product, yielding Equation 16 in the source: the mean square velocity is Planck's reduced constant times the natural frequency, divided by twice the mass. A similar calculation for the mean square fluctuation in acceleration yields Equation 17 in the source: the mean square acceleration is Planck's reduced constant times the cube of the natural frequency, divided by twice the mass.

With the above calculations in hand we are now in a position to characterize the ground state of the harmonic oscillator. First, the mean square fluctuation in position given by Equation 15 matches that obtained in the usual quantum mechanical treatment. Second, the mean square fluctuation in momentum, Equation 18 in the source — the mass squared times the mean square velocity, that is, the mass times Planck's reduced constant times the natural frequency, all over two — also matches that obtained from the standard quantum mechanical treatment. The harmonic oscillator's ground state energy, kinetic plus potential, is given by Equation 19 in the source: one half the mass times the mean square velocity, plus one half the mass times the squared natural frequency times the mean square position, which equals one half of Planck's reduced constant times the natural frequency — also in agreement with the quantum mechanical result.

Since the mean position and mean momentum of the stationary-state oscillator are zero, we also calculate the uncertainty relationship as Equation 20 in the source: the product of the root-mean-square position and the root-mean-square momentum equals the square root of the product of the mean square position and the mean square momentum, which is Planck's reduced constant over two — again in agreement with the known quantum mechanical result.

Now, in accordance with the argument being pursued here, we compare the average power being absorbed from the vacuum fluctuation distribution with that radiated due to accelerated motion, to determine their relative magnitudes. The power absorbed from the electric field due to the driving force — the charge times the zero-point electric field — is given by Equation 21 in the source as the average of the force dotted into the velocity, that is, the charge times the average of the zero-point field component along the motion multiplied by the velocity. With the zero-point field and the velocity given by Equations 3 and 5 respectively, the calculation carries through as in the derivation of the mean square position above to yield Equation 22 in the source: the average absorbed power equals the charge squared times Planck's reduced constant times the cube of the natural frequency, divided by twelve pi times the permittivity of free space, the mass and the cube of the speed of light.

The power radiated due to accelerated motion is given by the standard Larmor expression, Equation 23 in the source: the charge squared times the mean square acceleration, divided by six pi times the permittivity of free space and the cube of the speed of light. With substitution from Equation 17 and comparison with Equation 22, this yields Equation 24 in the source: the average radiated power equals the average absorbed power.

Thus we find that the ground-state parameters of the quantum mechanical harmonic oscillator can be accounted for on the basis of interaction between a harmonically bound point particle and the vacuum electromagnetic zero-point fluctuations. Specifically, the stationary ground state thus established derives from an average balance of power between that absorbed from the vacuum fluctuations and that radiated due to accelerated motion. It can be noted in passing that even as one asymptotically approaches the limit of vanishing charge — an uncharged oscillator — this outcome remains the same, as the charge squared cancels out in the relationship between radiated and absorbed power.

3. Generalized approach

Having derived the above relationship between absorbed and radiated powers for the harmonic oscillator's ground state, we now inquire as to whether this balance is specific to the harmonic oscillator by virtue of its simple linear restoring force, or can be extended to more general cases — a nonlinear oscillator, the hydrogen atom, a particle in a box, and so on.

We begin with the generalization of Equation 1, Equation 25 in the source: the mass times the acceleration equals the mass times the damping parameter times the third time derivative of position, plus the charge times the sum of the zero-point electric field and the cross product of the velocity with the zero-point magnetic field, plus an external force. We assume that external force is minus the gradient of a potential, for a broad class of cases of interest, with the potential a time-independent confining potential.

Multiplication of Equation 25 by the velocity, taking into account the mathematical simplifications — the velocity dotted into the cross product of velocity with the magnetic field is identically zero; one half the time derivative of the squared velocity is the velocity dotted into the acceleration; the total time derivative of the potential reduces, for a time-independent potential, to the velocity dotted into its gradient — followed by collection of terms, leads to Equation 26 in the source: the mass times the damping parameter times the mean square acceleration, plus the average of the time derivative of the quantity one half the mass times the squared velocity plus the potential minus the mass times the damping parameter times the velocity dotted into the acceleration, equals the charge times the average of the velocity dotted into the zero-point electric field.

At this point we recall the earlier-quoted statement that derives from quantum theory: "The crucial role of the vacuum fluctuations emerges in the ground state of matter. The stability of the ground state (i.e., the fact that it does not radiate) is purely a quantum effect which is due to the vacuum fluctuations." In the stochastic electrodynamics development presented here we find its expression in positing that application of Equation 26 to represent the — by definition — stationary ground state perforce requires that the time-dependent second term on the left-hand side must be conjectured to vanish. Otherwise Equation 26 could not be taken to represent a stable, time-independent ground state. As a consequence the remaining terms constitute the condition that reveals itself to be that the average power radiated due to accelerated motion, Larmor radiation, is balanced by the average power absorbed from the vacuum fluctuations. Substituting the definition of the damping parameter from Equation 8 we obtain Equation 27 in the source: the charge squared divided by six pi times the permittivity of free space and the cube of the speed of light, multiplied by the mean square acceleration, equals the charge times the average of the zero-point electric field dotted into the velocity.

Thus the stationary ground state, although — from a classical viewpoint — involving accelerated charged-particle motion and hence possessing an associated Larmor radiation loss, is nonetheless observed to be overall radiationless in nature due to a compensatory absorption from the background electromagnetic vacuum zero-point fluctuations. The balance so obtained also accounts for the well-known fact that an oscillator or atom in its ground state does not on net absorb zero-point radiation and therefore remains in its ground state. Finally, we note that this general result is independent of the form of the time-independent confining potential and is thus applicable to a wide range of problems.

Footnote 1 to this section. For the trivial case of a perfectly circular orbit this term vanishes even before averaging, given that the kinetic and potential energies are constant and the velocity and acceleration vectors are orthogonal. Beyond that, stochastic electrodynamics modeling attempts involving lengthy numerical simulations for hydrogenic atoms to verify the vanishing of the time-dependent term on the left-hand side of Equation 26 have to date only been marginally successful, self-ionization of the atom often being the outcome — as in the simulations of Cole and Zou and of Nieuwenhuizen and Liska. Under consideration are the incorporation of additional factors such as expansion beyond the dipole approximation, spin-orbit coupling, relativistic effects and so forth; thus simulation studies remain a work in progress.

4. Concluding remarks

Addressed is the seeming paradox that even though quantum ground states typically involve charged particles in accelerated motions, such states are nonetheless observed to be radiationless in nature. Though this feature is overlooked in everyday application of quantum theory to ground-state description, nonetheless this remarkable property is worthy of some discussion and clarification. In detail, it is the recognition that ground-state atomic structures are not isolated entities in an empty background, but are perforce immersed in a background of vacuum fluctuations. With regard to the behavior of charged particles, the primary component of interest is that of the vacuum electromagnetic zero-point fluctuations. Atoms, and other quantum systems, therefore constitute open systems engaged in dynamic interactions with the underlying vacuum states. Specifically, the on net radiationless characteristic of the ground state is seen to derive from particle–vacuum interactions in which a dynamic equilibrium is established between radiation emission due to particle acceleration, and compensatory absorption from the zero-point fluctuations of the vacuum electromagnetic field. Thus, employing a stochastic electrodynamics approach, we have shown here in detail that, as argued by Fain, the vacuum field is formally necessary for the stability of atomic structures, and this underlying principle therefore constitutes an important feature of quantum ground states.

Footnote 2 to this section. Further development of the foundational nature of the radiationless state for quantum mechanics in general can be found in the 2015 compendium by de la Peña, Cetto and Valdés-Hernández, which is an update to the earlier material. In this later work additional constraints on the stochastic electrodynamics approach are incorporated to match more closely the requirements of quantum electrodynamics, such as detailed balance of energy. In the updated approach, labeled linear stochastic electrodynamics, it remains the case that "the ZPF is seen as the source of the quantum behavior of matter."

Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author and the source, provide a link to the Creative Commons license, and indicate if changes were made. © The Author 2015.

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Harold E. Puthoff, Institute for Advanced Studies at Austin, 11855 Research Blvd, Austin, Texas. Published as Quantum Studies: Mathematics and Foundations 3, pages 5 to 10 (2016), doi.org/10.1007/s40509-015-0055-5, open access under CC BY 4.0.

(Running heads, page numbers and reference-number markers have been dropped, cited works are named in the prose instead, and display equations are rendered in words keyed to their source equation numbers; the sixteen references and the typeset equations are at the source.)

(On this site: the 1987 paper this one generalises, Puthoff’s zero-point-fluctuation-determined ground state of hydrogen, is at /library/stm-58f597c6df. The numerical simulations named in footnote 1 are Cole and Zou at /library/stm-e8f166a3ac and Nieuwenhuizen and Liska at /library/stm-9e011ff7d1; Nieuwenhuizen’s later work on the same problem is at /library/stm-fca91e355a and /library/stm-9ebf7acf49, and his regularisation of the harmonic oscillator treated in Section 2 is at /library/stm-b98f6364e3. Boyer’s founding papers on stochastic electrodynamics are at /library/stm-8c24f62160 and /library/stm-2d2d42369d, with Marshall’s earlier random electrodynamics at /library/stm-7faa238629; the de la Peña and Cetto programme named in footnote 2 is at /library/stm-bddbc3921c, /library/stm-d7a8b2ce17 and /library/stm-9409ed6eff. Puthoff’s companion papers taking the same balance into inertia, gravity and propulsion are at /library/stm-0f2b09effd, /library/stm-1ec4832b74, /library/stm-a62b2e761c and /library/stm-5cf7ebb6a4. Cole’s thermodynamic case for moving energy between matter and the field is at /library/stm-e61f12f673, and Moddel and Dmitriyeva’s assessment of extraction schemes built on this same picture is at /library/stm-d141795afd.)