Simultaneous Modes and Non-Radiating Atoms
1. Modes, Submodes and the Classical Paradigm.
Imagine that an uncharged particle of rest mass mo is subject to a central force that (a) acts perpendicularly to the particle’s velocity, and (b) has magnitude
According to Newton’s 2nd law (adjusted for the dependence of inertial mass on speed), the particle travels in a plane circular path of radius R and at constant speed v. To the extent the orbital plane could have any of an infinite number of orientations, it might be said that the system can access an infinite number of modes.
Let us define "The Classical Paradigm" as follows: At any instant t, a particle coincides with one and only one point in any frame of reference. In brief, according to the classical paradigm r(t) is unique in the chosen frame. A corollary would then be that, at any given moment, the time derivatives are also unique. For example, the particle cannot have more than one velocity at any given instant in one’s chosen frame, etc. To the extent we can observe r and v at any moment, we can both (a) justify the classical paradigm, and (b) determine the mode in any given case. The latter of course assumes that we do not perturb the system when we observe it, thereby knocking the system into another mode.
But the idea that we can observe the particle’s position, etc., without perturbing things does not alone guarantee that we can determine the mode precisely. For every microscopist knows that we cannot precisely determine r(t), even when mo is arbitrarily large. The precision of our knowledge of r(t) is limited (a) by the time it takes to make an observation, and (b) by the wavelength of the illumination (for example) that we use to "see" the particle. In brief, there is inevitably some small uncertainty in our knowledge of r and v at any particular moment. All of which means that we cannot know precisely what mode the orbiting particle occupies.
In macroscopic cases such considerations are ameliorated by making many consecutive observations (or even by observing the particle "continuously"). And long before physicists began to seriously investigate atoms, the nominal modes of the planets had been determined, corroborating the classical force laws and the laws of mechanics with a precision leaving little to be desired.
For future purposes it will be instructive to point out that we can divide each mode into an infinite number of submodes, each identified by a point on the mode’s circular path. For example, if the particle is at (R,0,0) at t=0, it might be said to occupy submode 0. If it is at (R-dx,dy,0) at t=0 it might be said to occupy submode 1, etc. Here again, according to the classical paradigm, the particle occupies one and only one mode and one and only one submode in any given instance. And here again, at least in macroscopic cases, one expects that the mode and submode can be quite precisely determined through careful observation(s).
2. Microscopic Systems.
Let us say that mo and R in Eq. 1_1 are both very small (say those of an electron and a Hydrogen atom). mo is so small that, barring near-light values of v, any attempt to determine r at any given instant will significantly modify the effect of F.
The situation is exacerbated if R is on the order of half a wavelength of the "illumination" being used. For the lesson of microscopy is that under these circumstances the particle could be anywhere within the system’s "volume." In effect it is impossible to determine which mode and submode the system occupies at any moment.
The consequences of this limitation may be profound, depending upon whether or not one believes that the classical paradigm applies even when it cannot be justified by observation. There are some who insist that r is unique at any given instant even if we cannot determine with any precision what it (and v) might be. Referencing Eq. 1_1, there would then be a unique angular momentum (L). Any attempt to determine the mode, and hence L, may change L. But (the belief goes) L is nonetheless unique.
There are others (this writer included) who suggest that the entire classical paradigm is only a mindset acquired when observing macroscopic systems. To be sure, in such macroscopic cases there may be ample evidence to corroborate the paradigm (at least to some acceptable precision). But it is presumptuous to insist that the paradigm also applies in microscopic cases, where it cannot be empirically justified.
But what should the paradigm be in any given microscopic case, if the classical paradigm does not apply? What makes sense if the laws of mechanics still apply, but the system actually does not occupy a unique mode? To the extent every mode is equally probable, the suggested answer is that it occupies every mode simultaneously! The consequences of such an idea may not be trivial. For example, with all modes simultaneously occupied, any given instance of the system might have zero angular momentum!
3. When the Particle is Charged.
A fundamental objection to ideas suggested by Bohr (and later generalized by Wilson and Sommerfeld) was that any bound, charged particle (e.g. an electron) should emit radiant energy … a theoretical result not in agreement with real-world, ground state atoms. But atoms are microscopic systems. And if we can shed our macroscopically acquired prejudice that the classical paradigm applies in microscopic systems, then it is not difficult to reconcile this non-radiating result with Maxwellian theory.
In order to appreciate this more fully, let us extend our generalization and grant not only that every possible mode may be occupied, but that every submode in any given mode may also be occupied. Any particular mode corresponds to the electron orbiting around a particular circular path. But if we grant that (at any instant) every point on one of these paths is equally probable, and if we then theorize that every submode is simultaneously occupied, then we can model the mode as a spinning ring of charge. And although particles theoretically radiate when they go in a circle, spinning rings of charge do not!
More generally, any microscopic orbit can be modeled as a circulating thread of charge. To the extent dB/dt=0 in every such case, we can expect zero radiation.
Modeling a simultaneously occupied, complete set of submodes in the above-described fashion is tantamount to theorizing that, in sufficiently cramped quarters, particles of charge (e.g. electrons) smear out into distributions of charge, with finite charge densities everywhere. Indeed this is the way Maxwell originally envisioned charge. It was only later that experimental results obtained by Townsend and others indicated that charge is particulate (at least when it is unbound).
What we end up with, then, is a distributed/particulate duality … particulate in unconfined spaces but distributed within the confines of microscopic volumes. At the root of this duality is the essentially justified acceptance of the classical paradigm in macroscopic cases, and its rejection as an unjustified prejudice in microscopic cases.
4. Multiple Energy States.
One of the fundamental premises of quantum theory is that the energies of microscopic systems can assume only discrete values when the systems are non-radiating. The question arises, can a given atom in a macroscopic sample simultaneously occupy multiple energy states (as well as multiple modes and submodes at any given energy)? The answer is, theoretically "Yes," and the weighting factors for the different energies are a function of the sample’s ambient temperature. Absorption of energy from the sample's environment skews the weights to higher energies, and emission has the opposite effect. In a solid whose temperature is constant and above absolute zero, the reality is theoretically dynamic, with lattice atoms exchanging energy quanta among one another. Here again one must not assume that a given atom is, at any instant, at any particular energy. The idea is that all the atoms in a given sample simultaneously occupy fractions of the set of accessible energies.
In conclusion, the point of the present article is that the classical paradigm, ‘though more or less justified in macroscopic observations, should not be assumed to apply in microscopic systems (where it cannot be observed to be true). It may require a bit of mental stretching to admit that all accessible modes and submodes may simultaneously be occupied in the case of a ground state atom (for example). The reward of such an adjustment, however, may be a reconciliation of Maxwellian theory with the empirical fact that ground state atoms do not emit radiation. It is essentially the novel feature that the theories of Heisenberg and Schroedinger added to the "old" quantum theory of Bohr et al. In deference to Schroedinger’s wave mechanics, another name for "distributed/particulate duality" is "wave/particle duality," although the distributed charge's density is actually proportional to the "square" of the wave function.