Bohr's Correspondence principle tells us that in the limit of large quantum numbers, quantum physics will agree with classical physics. For example, the simple Rutherford atom model of hydrogen, where a point charge electron orbits a point charge proton in a classical Keplerian orbit, will increasingly agree with observation, in terms of rate of radiative decay and frequency of transition radiation, as the electron orbital energy and angular momentum is raised. (The study of so-called Rydberg atoms, that is, atoms where the outer electron is excited to a high energy level compared to the rest, confirms this.) The frequency of the electromagnetic radiation becomes in the limit of large quantum number simply the frequency of revolution of the electron in its classical orbit. The rate of energy loss is also calculable from the radiation intensity predicted for an accelerating point charge according to classical electrodynamics.

At lower energy levels, and apart from the issue of quantized energy levels, the classical model of radiative decay of the system of bound point charges deviates significantly from observation. There are at least two obvious differences from what the classical model predicts. The transition time between two defined energy levels is less than the classical model predicts, and the frequency of the radiation is inconsistent with the classical expectation. The classical expectation is that the radiation of decay between two energy levels will start at the orbital frequency of the higher energy level and end at the (higher) orbital frequency of the lower. Observation however shows that transition radiation is essentially monochromatic, with the frequency predictable based on simply the energy difference between the levels. These seemingly inexplicable differences with the expectations of classical physics led to some despair at the time of their discovery. The initial triumph of quantum theory, even the early quantum theory of Bohr and Sommerfeld, and later the modern version developed particularly by Heisenberg, Schroedinger, and Dirac, was its ability to accurately describe atomic spectra including these features.

Before abandoning the classical picture as hopeless, however, it may be worthwhile to consider what are the implications to it of the existence of the intrinsic spin as a property of elementary particles. Although ubiquitously known today, the existence of intrinsic spin was not commonly appreciated until approximately coincidentally with the advent of modern quantum theory separately by Heisenberg and Schroedinger in 1926. Neither Heisenberg's nor Schroedinger's quantum theory formulations made initial use of the spin, although it was incorporated into them in an ad hoc fashion in short order. Dirac then put it on a firm foundation in 1928. In the excitement of the emergence of a broadly working and essentially non-classical quantum theory, however, it seems that not much attention was paid to whether the intrinsic spin might be the missing ingredient that could make classical physics applicable to atomic scale systems.

It's worth noting that the effect of intrinsic spin is quite ubiquitous in the quantum world. The coupling between orbital and spin angular momentum has an effect on atomic emission spectra that was noted early on and eventually led to postulation of the existence of intrinsic spin as a new particle property. It can be further observed that the onset of significant spin-orbit coupling effects is coincident with the onset of quantum behavior. This alone should provide strong motivation for investigating what are the implications in classical physics of the existence of intrinsic spin.

The existence of intrinsic spin greatly enriches the picture of radiative decay according to classical electrodynamics. As outlined in the previous post, this richness is most apparent when there are multiple particles with both significant intrinsic spin and significant intrinsic magnetic moment, the simplest example of which is the Positronium atom consisting of simply an electron and an anti-electron (i.e., a positron) bound together by Coulomb (i.e., electrostatic) attraction. Is it at all plausible that the classically-predicted radiative decay of such a system could be similar to that observed? I believe the answer is yes. There is clearly a new radiative mechanism predicted that may account for the higher observed rate of decay of quantum systems compared to classical systems of point charges with charge but without spin. Whether this mechanism can properly account for observed monochromaticity of atomic transition radiation remains to be seen, but also seems plausible for reasons to be elaborated below.

The additional mechanism of radiation present in classical systems where the particles have intrinsic spin in addition to electric charge arises due to several effects. First, charged particles with intrinsic spin are expected according to classical physics to have magnetic dipole moments proportionately to their spin angular momentum magnitude and charge-to-mass ratio. These intrinsic magnetic moments are of course recognized to exist. (The fact that they are larger than expected by about a factor of two for electrons need not concern us here. The magnitude of the spin angular momentum and its associated magnetic moment larger than classically expected, can simply be accepted as given quantities in the quasi-classical theory that incorporates intrinsic spin.) Second, a moving magnetic dipole necessarily acquires an electric dipole moment by the laws of relativity. This fact is important because electric dipoles radiate much more efficiently than magnetic dipoles. (Radiative decay due to the magnetic dipole radiation at the expected acceleration magnitude of the magnetic dipole moment would take much longer than observed.) Third, a classical system of electrically bound spinning particles will exhibit motions of the spin axes of the particles due to the rapidly time-varying electromagnetic fields experienced by the particles. These motions lead directly to an additional and richly complex radiative mechanism to the simple mechanisms of an accelerated point charge of the Rutherford model. Furthermore, this mechanism will become insignificant as the spin-orbit coupling becomes insignificant in the limit of large quantum numbers, exactly as required for consistency with the Correspondence principle.

Although strictly speaking intrinsic spin is not an allowed particle property according to classical electrodynamics, if it is included on an ad hoc basis classical electrodynamics provides adequate tools for assessing its behavior in bound systems. In particular, the orientational motion of the spin in an electromagnetic field is described by the Thomas or covariant but equivalent Bargmann-Michel-Telegdi (BMT) equations. To describe the radition from mutually-bound spinning particles, it is merely necessary to determine the time-varying electromagnetic field encountered by the particles, find the resulting spin orientational motion via the Thomas or BMT equations, translate this motion into an equation of motion of the magnetic and electric dipole moments due to the spins, and add these to the dipole moments due to the motion of the charges. The raditation intensity as a function of time is then determined straightforwardly from acceleration (that is, the second derivative with respect to time) of the total dipole moment of the atomic system.

The equation for the magnitude of the acceleration of the total electric dipole moment of positronium is quite complicated, even in the simplest possible case where the particles are mutually circularly orbiting. This comparative complication relative to the simple equation of electric dipole moment acceleration for the Rutherford atom is what allows the possiblility for the emergence of a justification for quantum behavior. The equation for the vector acceleration of the total atomic dipole moment is fairly complicated to begin with, having orbital-frequency contributions due to both charge separtion between the electron and positron, and orbital frequency orientational motions of the spins which have acquired an associated electric dipole moment through the orbital motion of the spin-associated intrinsic magnetic moment. Taking the magnitude of the electric dipole acceleration then gives rise to cross-product terms that mix the electric dipole moments due to charge separation and intrinsic spin together. The question of whether this description can predict the atomic spectrum of Positonium then becomes simply one of analysis. Does the total electric dipole moment acceleration magnitude vanish for some conditions? Is the frequency of oscillation effectively a constant under some conditions? I believe the answers to these questions are worth pursuing. Also, it should be noted, the radiation intensity equation obtained in this way contains Planck's constant in a natural manner, carried in by the magnitude of the intrinsic spin. Thus, any vanishing of the radiation intensity will be expressed as a function of the Planck constant, similarly to quantum mechanics.

I plan to describe my ongoing analysis of the quasiclassical radiative properties of the Positronium atom in future posts.

Saturday squirrels

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