On the Stability of Classical Orbits of the Hydrogen Ground State in Stochastic Electrodynamics

Abstract

De la Peña 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics connect to a stable situation, where the electron neither collapses onto the nucleus nor gets expelled from the atom. Although the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Peña-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an inverse-square potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.

Keywords: Stochastic Electrodynamics; hydrogen ground state; stability criterion

Author and affiliations. Theodorus M. Nieuwenhuizen (Institute for Theoretical Physics, Amsterdam, The Netherlands; International Institute of Physics, UFRG, Natal-RN, Brazil). Entropy 2016, 18, 135. Academic editors: Gregg Jaeger and Andrei Khrennikov. Received 19 February 2016; accepted 31 March 2016; published 13 April 2016.

1. Introduction

Stochastic Electrodynamics (SED) is a subquantum theory that considers the quantum vacuum as a true physical vacuum with its zero-point modes being physical electromagnetic modes. By construction, the vacuum is filled with classical electromagnetic modes, each carrying an energy of one half of the reduced Planck constant multiplied by the mode frequency, where Planck's constant is a constant of nature characterising the energy stored in these modes. The task is then to show that SED explains all quantum behaviour of matter at the statistical level. This has been argued to occur on general grounds. SED is not ruled out by Bell's theorem, since that has an irreparable fatality, the contextuality loophole.

Testing the working of SED in special cases is needed to get trust in the theory. Since harmonic oscillators perform well, several phenomena are explained: van der Waals forces, the logarithm of the Lamb shift between the hydrogen 1s and 2p states, the Casimir effect, the Unruh effect. A natural next case is the hydrogen ground state. de la Peña and Puthoff demonstrate that circular orbits lead to a stable atom: they do not fall onto the centre and they do not self-ionise. The SED theory has been tested numerically on the hydrogen ground state in a two-dimensional approximation where the orbit remains in its initial plane. Cole and Zou 2003 observed an encouraging agreement with the result from the quantum ground state wave function. New simulations have been carried out recently by our team in a three-dimensional approach, benefiting from a decade of progress in computational and programming power and employing a few analytical tricks to make the problem tractable in three dimensions. However, it was observed that in all runs and all attempts to model the system, there occurred self-ionisation. The latter fact agrees with a theoretical prediction. Boyer puts forward that the problem may lie in relativistic effects, but taking into account relativistic corrections in our computer code did not change much and in particular did not cure the self-ionisation.

Faced with the renewed finding of self-ionisation, we search for a theoretical underpinning for it and, reconsidering the stability argument of de la Peña and Puthoff, we extend it to include eccentricity. We derive the gain and loss terms, averaged over the stochastic force and averaged over a full period, in Section 2. In order to study a more general problem with possibly a stable state, we extend in Section 3 the problem with an attractive or repulsive inverse-square potential. We close with a discussion. In an appendix we consider the case where both masses are finite, such as the positronium.

2. On the Stability of the Hydrogen Ground State

Equations 1 to 9, described here. The nonrelativistic equation of motion of an electron of charge minus e around an ion of charge Z times e is written in a dimensionless form, with distances measured in Bohr radii and times in the Bohr time. It carries three terms beyond the Coulomb attraction: the fluctuating electromagnetic field that lies at the heart of SED, multiplied by a small coupling constant which is two thirds of the fine structure constant to the three-halves power multiplied by the nuclear charge number; and a damping term proportional to the square of that coupling and to the third time derivative of position. Because that coupling is of order the nuclear charge number divided by two thousand, one may consider a time window during which the orbit remains basically unperturbed and in a plane. The correlator of the field is a fourth-power kernel in the time difference, regularised by a cutoff of the order of the Compton time, which is taken to zero at the end of the calculation.

The unperturbed orbit is the Kepler ellipse written in the usual way through the angular momentum, the eccentricity and the angle between the long axis and the x-axis, all three conserved in the absence of stochastic fields and damping. The orbital energy is minus one half of the square of a parameter written as k, and the eccentricity, that parameter and the angular momentum are tied together by a single relation. The angle around the orbit is then coded through an auxiliary angle in the standard parametric form, so that the position, the time and the radial velocity along the unperturbed orbit are all expressed through that one auxiliary angle, and the true angle is recovered by an arctangent formula with a floor-function branch counter.

2.1. Perturbations Around the Kepler Orbit

The energy changes due to the stochastic field and due to radiation. Due to the small value of the fine structure constant these effects can be calculated separately. In a perturbation expansion to second order in the coupling, the rate of energy change due to the stochastic field follows from expanding the position in powers of the coupling and taking the scalar product of the field with the velocity. With the leading term vanishing on the average, there remains a statistically averaged energy gain from the field, second order in the coupling, given by the average of the field against the first-order correction to the velocity.

We shall suppose that the stochastic fields and the damping start to act at an initial time, which is later taken to minus infinity. The solution for the first-order perturbation has the form of a time integral of a Green's function against the field. Hence the average effect by the field, for times during which the Kepler orbit is not disturbed much, reads as a time integral of the three-dimensional trace of the time-derivative of the Green's function against the field correlator.

2.2. Perturbations of the Orbit

We solve the perturbed Kepler dynamics in the rotating frame where the orbit is decomposed on the two rotating eigenvectors and the fixed perpendicular vector. The field then has three components on that comoving basis, and the linearised equation of motion becomes three coupled equations: two in the plane, coupling the radial and tangential displacements through the orbital rotation, and one transverse.

Equations 15 to 19, described here. Six homogeneous solutions of that system are constructed analytically. The first is the orbital velocity itself. Another has vanishing radial component and a tangential component proportional to the radius. Two further in-plane solutions are more intricate but can be constructed in closed form, one of them containing oscillations around a term linear in time. For the transverse displacement one finds the two solutions given by the components of the orbit on the non-rotating basis. The Green's function on the rotating basis is then assembled from these six solutions as an antisymmetrised sum of products, divided by the eccentricity and the cube of the energy parameter, and satisfies the required conditions that it vanish at equal times and that its time derivative there equal the identity. The transverse element comes from the last pair of solutions and takes a compact form in the two auxiliary angles. It is easily verified that the equal-time derivative is a matrix-valued Wronskian, that is, a constant matrix, indeed equal to the identity.

2.3. Statistical Rate of Energy Change From the Stochastic Field

Equations 20 to 31, described here. The field contribution to the energy change is a time integral of the trace of the Green's function derivative, with three subtracted, against the field correlator; the subtracted term does not contribute to the integral. Expanding the transverse and the in-plane parts near equal times shows that their second and third order terms in the time difference are potentially dangerous for the convergence of the integral, but are exactly opposite, so that the combination is of fourth order in the time difference. This behaviour assures that the cutoff can be taken to zero and the integral behaves well, as it should, since no subtle short-time effects are expected for the energy absorbed from the field. The combined effect takes a compact form as a ratio whose numerator is built from one antisymmetric and one symmetric trigonometric function of the two auxiliary angles.

The energy gain from the field, averaged over one period, is then written as one half of the squared coupling, multiplied by the ninth power of the energy parameter, divided by the sixth power of the eccentricity factor, multiplied by one plus the square of the eccentricity, multiplied by a gain function of the eccentricity factor. That gain function is quite smooth. For circular orbits it ends at one half. Its maximal value, reached in the limit of extreme eccentricity, is obtained by rescaling the two auxiliary angles, and after performing one of the two integrals analytically it evaluates to sixteen divided by five pi times the square root of three, that is, 0.588084 — a result not far above the circular-orbit value of one half.

Radiative Energy Loss

Equations 32 and 33, described here. The loss terms produce a rate of energy loss averaged over one orbit which equals minus the squared coupling multiplied by the orbital average of the inverse fourth power of the radius; in the averaging over a full period a total derivative could be omitted. Carrying out that average over the Kepler ellipse gives a closed expression proportional to the eighth power of the energy parameter and to three minus the square of the eccentricity factor, divided by twice its fifth power.

Equations 34 to 37, described here. Combining gain and loss terms leads to a total energy change proportional to the eighth power of the energy parameter, divided by twice the sixth power of the eccentricity factor, multiplied by two plus the square of the eccentricity, multiplied by the difference between the energy parameter times the gain function and the eccentricity factor.

For spherical orbits there is the explicit result, first derived qualitatively by de la Peña and Puthoff: the total energy change is the squared coupling times the eighth power of the energy parameter times one half of that parameter minus one. It exhibits stability: for large negative energy the energy gain is positive on the average, preventing collapse on the nucleus. On the other hand, when the energy is small and negative, its average change is negative, preventing self-ionisation.

But in the limit of extreme eccentricity at fixed angular momentum, the total energy change becomes proportional to the difference between the extreme-eccentricity value of the gain function and the angular momentum itself. Per period in that limit this implies a fixed amount of energy gained. In words: each revolution produces, on the average, a finite amount of energy, so that self-ionisation happens for orbits that have achieved a small energy parameter and an angular momentum below the extreme-eccentricity value of the gain function. The pericentre of the orbit lies at approximately half the square of the angular momentum. The critical angular momentum value 0.588057 is not particularly small, and neither is the critical pericentre, 0.172921 Bohr radii, which is 23.7 times the fine structure constant.

Figure 1. Distribution of the dimensionless angular momentum L (in units of the reduced Planck constant) from the numerical simulation of the Stochastic Electrodynamics reported in reference 8. Not much weight lies below L = 0.588, confirming that when such a value is reached at near-zero energy, self-ionisation may occur rapidly and the run is ended. Full curve: the distribution of L from the conjecture for the would-be stable ground state distribution, Equation 68 in the limit of vanishing dipolar coupling.

We are faced with the finding that the de la Peña-Puthoff stability argument based on circular orbits does not hold for all orbits, in particular not for very eccentric ones. This is in full accord with our recent numerical simulations. There, for short times, orbits with reasonable statistics were observed, sometimes going close to the nucleus or near zero energy, followed by recovery towards less extreme orbits. But at some moment no recovery from a near-zero energy orbit was observed, with self-ionisation as a result. This characteristic is confirmed by the distribution of the dimensionless angular momentum obtained from these simulations: in Figure 1, hardly any weight of orbits lies below 0.588.

3. Adding an Inverse-Square Potential

In the hope of finding some stable behaviour, we extend the problem without spoiling its analytical solvability. We consider the presence of a "radially-directed dipolar force", stemming from a potential equal to minus a coupling d divided by twice the square of the radius, which has the same form as the angular momentum part of the kinetic energy. Such a term may arise from a dipolar force between electron and nucleus, when the deformation of the nucleus, caused by the electron, is always aligned with the vector between them.

3.1. Analysing the Orbits

Equations 38 to 42, described here. The unperturbed problem now has a potential with both the Coulomb term and the new inverse-square term, and the dynamics adds the same random force and damping as before. For an unperturbed orbit the radial parametrisation through the auxiliary angle is unchanged, but the presence of the new coupling changes the relations among eccentricity, angular momentum and energy: an effective angular momentum appears, and the eccentricity factor and the scaled angular momentum are no longer equal.

In the repulsive case the coupling is negative, the physical ranges of the eccentricity and the eccentricity factor are bounded, and the condition that the scaled coupling be less than one in magnitude implies that the orbital energy is bounded from below.

For the attractive values of the dipolar force, the eccentricity and the eccentricity factor range between zero and one, while the scaled angular momentum takes values in a shifted band. However, now there is a second family of orbits having angular momentum below the square root of the coupling, where the square of the effective angular momentum is negative and the eccentricity exceeds one. While such orbits are hyperbolic and unbound in the pure Coulomb case, they are bound when the dipolar coupling is attractive: these low-angular-momentum orbits spiral into and then out of the nucleus and need to be described within a relativistic framework. This property may explain the Darwin term, a delta-function term in the relativistic corrections to the hydrogen ground state, related to our problem with a coupling equal to the square of the fine structure constant, as an effect of central spiralling.

The relation between the angular velocity and the angular momentum now brings a rotation angle multiplied by a factor equal to the angular momentum divided by the effective angular momentum. One period in the auxiliary angle still takes the same time, but it involves that many turns in the true angle, a number which is in general non-integer. For attractive coupling each turn takes a shorter time, so although the radial solution remains, the orbit rotates faster; for repulsive coupling it rotates slower. When the ratio differs from one, the sines and cosines of the angle difference do not reduce to the compact forms of the pure Coulomb case.

3.2. Perturbations of the Orbit · 3.3. Statistical Rate of Energy Change from the Stochastic Field

The construction of Section 2 is carried through for the modified potential: six homogeneous solutions, the Green's function assembled from them, and the trace integrated against the field correlator to give the averaged energy gain. The expressions are longer than in the pure Coulomb case and are given in full at the source.

3.4. Circular Orbits

Equations 52 and 53, described here. At zero eccentricity the problem is regular. For repulsive dipolar coupling there is a maximal value of the energy parameter, equal to the inverse square root of the magnitude of the coupling, and the total energy change is a polynomial in the scaled angular momentum which prevents the result passing below the lowest energy coded by that maximal value. For small energy parameter the loss term dominates, exhibiting de la Peña-Puthoff stability of the atom.

For attractive coupling the total energy change is again explicit. At small energy parameter it always shows stability. At large energy parameter the stability condition demands that the coupling be less than one quarter, a condition met curiously already in the quantum approach. So the dipole force does not essentially modify the stability of circular orbits.

In conclusion, for circular orbits the hydrogen atom retains its de la Peña-Puthoff stability in the presence of the new potential.

3.5. Very Eccentric Orbits

The pure hydrogen case taught us that the remaining interest lies in the limit of very eccentric orbits with energy close to zero, where the possibility of self-ionisation looms. In that limit the loss term scales so as to give a finite contribution per period, as before, expressed through a quartic polynomial in the ratio of angular momentum to effective angular momentum.

Equations 64 and 65, described here. Self-ionisation is likely prevented when a certain function of that ratio, built from the gain integral divided by the same quartic polynomial, stays below the square root of the magnitude of the dipolar coupling for all orbits, that is, for all relevant values of the ratio.

In the repulsive case the range for the ratio runs from zero to one. Since the function takes the value 5.99 at zero and vanishes at one, our statistical argument suggests a stable bound state above a finite threshold in the coupling. This threshold is finite, though rather large.

For attractive coupling the physical domain is the ratio squared at least one, or negative. Where the ratio squared is at least one we find that the function is increasing, and its asymptotic behaviour can be analysed. The limit of large ratio describes the orbits with lowest possible effective angular momentum, namely zero. Evaluating the leading and subleading terms — for the second-order correction a regularisation is needed, a subtraction of total derivatives — gives an expansion in inverse powers of the ratio whose leading term is one half. With this shape, the stability condition predicts that these orbits remain stable only at the extremal coupling of one quarter, but then the circular-orbit result predicts that spherical orbits sink to the centre, so no stable cases are found for attractive coupling. The role of orbits spiralling into and out of the centre is left as an open question.

3.6. Quantum Mechanics

Equations 66 and 67, described here. In a quantum approach one would introduce the angular momentum operator and an effective one, the latter taking eigenvalues shifted by the dipolar coupling, so that the effective orbital quantum number is a square-root expression which for the ground state imposes that the coupling not exceed one quarter. The nonrelativistic Schrödinger equation for the radial wave function then carries both the Coulomb term and the new one. For a coupling equal to the square of the fine structure constant times the square of the nuclear charge number, it has an analogy with the Schrödinger equation for the spinless relativistic electron in a hydrogen atom. Indeed, it produces the Dirac square-root formula for the eigenenergies, except that the total angular momentum operator for the spinning electron is reduced to the orbital one and hence its eigenvalues follow the orbital quantum number. The ground state wave function and the ground state energy both follow in closed form, the energy being minus one half divided by the square of one plus the effective ground state quantum number.

3.7. Classical Phase Space Density

Equations 68 to 75, described here. If a stationary ground state exists in the SED problem, it should be expressible as a function of the conserved quantities, more precisely as a function of the inverse energy and of the effective angular momentum. Since this task was worked out by us for the ground state and excited states of the relativistic hydrogen atom, we can adjust the approach here. A trial phase-space density is proposed in that form and its normalisation is fixed by carrying out the momentum-space integral, in which the effective angular momentum rather than the angular momentum itself enters. The resulting joint distribution of energy and of the scaled angular momentum is normalised to unity when taken over the full physical range.

Despite the setback for the SED program for vanishing and for all attractive dipolar coupling, it would be interesting to test this distribution for the repulsive regime beyond the threshold, where a stable ground state of the problem should occur.

4. Discussion

It was put forward by de la Peña 1980 and Puthoff 1987 that circular orbits lead to stability in the hydrogen problem of Stochastic Electrodynamics. This gave hope for stability of the full problem, supported by the 2003 Cole-Zou numerical simulations of the dynamics. Our own, recent simulations improved on these, benefitting from new computer power, new architectures and analytical tricks. However, it was found that self-ionisation always occurs. This is explained by the present analytical approach.

We have developed the de la Peña-Puthoff statistical theory, where the average energy gain from the field per period of the orbit is evaluated and compared with the average energy lost by radiation. Since both parts are defined by the unperturbed problem, the derivation is elegant and prone for study by students.

Our approach predicts indeed that with the electron and the nucleus modelled as point charges, self-ionisation takes place in the SED description of the hydrogen atom. Technically, the problem arises from orbits with nearly vanishing energy and moderately small angular momentum, below 0.5880 in units of the reduced Planck constant (we restore physical units). The perihelion then lies at distance 0.173 Bohr radii from the nucleus. The speed takes here its maximum of 0.024 times the speed of light, so the problem likely is insensitive to relativistic corrections, as we verified numerically. Indeed, when one looks at Figure 1, the problem is not only the self-ionisation below that angular momentum, but moreover that the whole distribution is different from what one would expect from a conjecture based on the shape of the quantum ground state wavefunction. Hence if the SED program can be saved, either the gain or the loss term, or both, need to have a different shape.

Next we have added the dipolar potential, which for a coupling equal to the square of the fine structure constant times the square of the nuclear charge number has connections with the relativistic hydrogen problem for a spinless electron. The problem remains exactly solvable. In the repulsive case, above a critical coupling of 28.6 in magnitude, we predict stability, which would be an interesting test for our numerical approach. In the attractive situation we always find instability, even without accounting for effects from orbits which spiral into and out of the origin in a finite time. Those orbits nevertheless offer in principle a connection with the Darwin term, a relativistic delta-function correction to the Hamiltonian for the hydrogen problem, so that for some scholars the SED theory will retain a magic spell.

Our results support the conclusions reached by a number of authors regarding the failure of the "old" approach in SED to account for atomic stability: atomic orbits are not simply classical orbits perturbed by the stochastic field. For a proper functioning of SED, an intricate equilibrium state of both the electron and the stochastic field seems to be needed.

Acknowledgments. It is a pleasure to thank Erik van Heusden and Matthew Liska for discussion, and the latter also for allowing publication of Figure 1.

Conflicts of Interest. The author declares no conflict of interest.

Appendix: The Positronium Problem in SED

Equations A1 to A6, described here. Starting from the standard expression for the electric field created by a moving point charge, written through the retarded separation vector, the retarded velocity and the retarded acceleration, the field is expanded at fixed observation point in a power series in velocity over the speed of light. Terms of second order in the inverse speed of light are dropped, because they are small relativistic corrections to the Coulomb force, but the radiation terms are kept and the radiation self-terms are added. The transverse field is the fluctuating SED field in the dipole approximation, where its spatial dependence is neglected. Written in terms of the centre-of-mass coordinate and the mutual coordinate, the two-body problem separates into an equation for the centre of mass driven by the total charge, and an equation for the relative coordinate driven by an effective charge built from the two masses and the two charges, with the reduced mass appearing on the left.

For hydrogen and positronium the charges are opposite, so the total charge vanishes, the effective charge is minus the elementary charge and the centre of mass stays fixed. This results in a relative-coordinate dynamics similar to that of one electron around an alkali ion, with its non-zero but negligible centre-of-mass motion.

In conclusion, the results of this paper hold also for positronium after replacing the electron mass by the reduced mass of one half of the electron mass, because the mutual electric fields contain relativistic corrections comparable to the self-damping terms.

(The fifteen references are at the source.)