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The Evolution of a Quantum Paradigm

G.R.Dixon, 10/26/2006

Authorís Note: This article provides a short historical review of the way we think about atoms and other "microscopic" systems. It concludes with a previously suggested step beyond the present ensemble/statistical view of YY*dV. A somewhat more mathematical treatment is provided in two earlier articles. About the only new idea here is the authorís bewilderment at deBroglieís choice of standing waves in the case of Bohr Hydrogen atom electron orbits. As with all articles featured on this site, feedback is welcome. I hope that all who read it are amused but not overly bemused. GRD.

1. The Classical Paradigm.

According to the Classical Paradigm, at any given instant every particle has a unique position, velocity, acceleration, etc. relative to any frame of reference. This premise is ubiquitous in classical physics, including Special Relativity. Indeed it might be what Einstein had in mind when he refused to believe that God plays dice with the universe. The paradigm was (and, for many, continues to be) essentially axiomatic, despite the fact that it has long been acknowledged that an arbitrarily massive particleís position cannot be determined with absolute precision. An important corollary to the kinematic aspects is the belief that, at least in principle, every system has a unique energy at any moment in time. Similar remarks apply to angular momentum.

2. The Bohr Hydrogen Atom.

Bohrís "solar system" model for the Hydrogen atom fully embraces the classical paradigm. A new feature in Bohrís model, however, is the ad hoc limitation of electron angular momenta to discrete values. An obvious objection to the model is the Maxwellian result that any charged particle, going in a circle, should constantly emit radiation. At best this objection was initially dismissed as a disagreement with electromagnetic theory that required further thought.

It should be acknowledged that Bohr evidently distanced himself from the classical paradigm later in his career. It would be difficult to understand his celebrated debates with Einstein, had this not been the case. The author recalls an exhibit many years ago at the Los Alamos National Laboratory, in which an older Bohr suggested a "vibrating droplet" (or something to that effect) model for a radiating atom, the droplet presumably being an atomic scale distribution of negative charge.

3. deBroglieís Particle Waves.

The first real break with the classical paradigm was deBroglieís suggestion that some sort of extended wave is associated with every particle. Among other things, deBroglieís formula relating particle momentum to an associated wavelength suggested an explanation for the Bohr angular momentum rule. For only orbits accommodating integral numbers of wavelengths would be nonzero. The idea was later lent support by Bornís suggestion that the waveís square constitutes the probability density of a particle being at (or being detectable at) a given point.

It is something of a mystery to this writer why deBroglie imagined counter-rotating waves, resulting in a standing wave. An obvious difficulty with standing waves is their alternating flat-lining and amplitude maximizing. Single wave trains, propagating around a closed loop in one direction or another, would also have to exist in integral wavelengths in order not to sum to zero. And single waves would not flat-line.

4. When Circling Charges Do Not Radiate.

The deBroglie postulates did not mitigate the Maxwellian objection that orbiting electrons should radiate. But one of the fascinating consequences of Maxwellian theory is that a spinning ring of charge (with constant angular velocity) does not radiate. Alone and isolated, any charge increment in the ring would radiate. But joined end-to-end in a ring, the entire set of charge increments does not radiate. (Similar remarks apply to certain 3-dimensional distributions, such as pulsating spherical shells of charge.)

5. A Suggested Point/Distributed Duality.

Perhaps influenced by such considerations, certain theorists (including a young Feynman) considered the possibility that, within the minuscule confines of an atom, bound electrons lose their particulate nature and disperse into finite distributions of charge. This seems to be what Bohr had in mind with his vibrating droplet model.

A working formula for the charge density in such cases utilized the quantum wave function solutions of the Schroedinger equation. Taking into account Bornís suggestion, that Y(r,t)Y*(r,t)dV is the probability of a particle being in the infinitesimal volume centered on r at time t, some theorists suggested that the formula for the negative charge density within an atom (at points external to the nucleus) is qYY*, where q is the total amount of negative charge in the atom.

6. Ensembles and Statistics.

At this stage in the theoryís evolution it is fair to say that belief in the classical paradigm is still broadly subscribed to (although Feynman admonished his students that atomic behaviors are like nothing theyíd ever seen in ordinary life). In effect, Bornís meaning of YY*dV takes on a statistical flair. The idea is that, given an arbitrarily large collection of atoms (or an ensemble), all with the same energy, YY*dV specifies the fraction in which an electron would be found in dV at any given instant (with each nucleus defining an Origin). In brief the belief that, in any particular atom the electron must be at some unique point, has not readily been abandoned.

An objection to the above-described collection is that an ensemble of mono-energetic atoms might be realizable only with great difficulty. (See discussions of Bose Einstein Condensates, superfluidity, etc. on the Internet.) Virtually all real-world collections of atoms, at temperatures greater than absolute zero, contain atoms at many energies. And the situation is dynamic, with the collection constantly exchanging energy quanta with the environment, and with the atoms in the collection doing the same among themselves.

7. A Quantum Paradigm.

At this point it seems advisable to revisit the classical paradigm in its entirety. Since any particular Hydrogen atomís lone electron could be virtually anywhere in the atomís volume, it doesnít seem unreasonable to conclude (a la the elder Bohr, the young Feynman, etc.) that the electron is simultaneously everywhere in the atom. Such a conclusion is consistent with the notion that, in such confined spaces, the electronís particulate nature gives way to a distributed one.

But if a bound electronís position in a particular atom is impossible to know, a priori, then might the same not be true of its velocity and ultimately of the atomís total energy? Should we perhaps make a clean break with the classical paradigm altogether, and admit the possibility that, a priori, a given atom simultaneously occupies an entire set of energies, with the occupancy weighting factors being a function of the ambient temperature?

8. Bohrís Correspondence Principle.

Early in the game Bohr enunciated his Correspondence Principle, which stated that quantum theory must be consistent with all macroscopic phenomena (e.g. colliding billiard balls). The implication is that everything we observe in the macroscopic world must be consistent with quantum theory. To the extent the classical paradigm is irrelevant in the study of atoms, we must admit the possibility that every macroscopic system (e.g. our solar system) simultaneously occupies multiple energy states!

But surely, it might be argued, we can quite precisely determine particular energy states in macroscopic cases. In response to this objection, it need only be pointed out that, in theory and in such cases, we observe very tightly clustered sets of energy states, with the difference between adjacent energies being too small to measure. True enough, in such macroscopic cases we are usually justified in employing the "cleaner" classical paradigm. But the proof that multiple, simultaneous energy states might constitute the bed rock reality becomes more evident as we transition from the observation of macroscopic systems to observing microscopic ones. In brief, the classical paradigm might in the final analysis be an illusion, rooted in our observation of macroscopic phenomena.

Such a total abandonment of the classical paradigm might constitute the final step in understanding the physics of atoms. In essence we should not assume that, in any multi-atom sample, each constituent atom has some particular energy (even though the collection contains atoms at many different energies). Rather we should assume that every atom in the sample simultaneously occupies multiple energy states. The weighting factors would be a function of the ambient temperature, and would apply to the entire sample.

Although such a quantum paradigm might at first seem nonsensical to our macroscopically acquired notions of energy uniqueness, etc., it may in the final analysis constitute the reality of macroscopic, as well as of microscopic, systems.

9. Y(r,t) When T>0.

An important ramification of the multiple energy state occupancy idea is that Y, in any given atom, is the weighted sum of the Schroedinger equation solutions for each energy in the set. To the extent the weighting factors apply to the entire sample, every atom has the same net wave function. a priori we can only assume that every atom occupies a set of energies.

It is an interesting result that, when two or more solutions for Y (corresponding to two or more energy levels) are added, then the resultant wave function generally has both a time-independent and a time-dependent part. Regarding the latter, the implication is that, at any given point (or more generally at any distance from the nucleus), the charge density varies in time. This alone does not dictate that the atom emit (or absorb) radiant energy. It is only when the spherical symmetry of r(r,t) is perturbed that, from the perspective of points adequately removed from the atom, the nucleus and bound electron behave like an oscillating dipole, emitting/absorbing radiant energy as they do so.

To the extent every atom in a sample occupies multiple energy levels identically, it is a contradiction to conclude that a given atomís energy decreases/increases when radiant energy is emitted/absorbed. Unless we "monitor" (and thereby interact with) the atoms in a sample, we cannot say a priori which atoms emit and which atoms absorb energy quanta during any short interval of time. And if we do determine that a particular atom has emitted or absorbed a quantum of energy, we necessarily interact with it, injecting/extracting extraneous quanta in indeterminate amounts.

The situation is certainly different from a macroscopic tub and bucket, where we have exact knowledge of how many buckets of water we have added to/removed from the tub (not to mention how many buckets the tub contains at any moment). In the atomic analogy the tub simultaneously contains 1 bucket, 2 buckets, etc., with different weighting factors. Such a prospect may seem nonsensical to a mind steeped in the classical paradigm. But as Feynman advised his students, atoms appear to behave like nothing we observe in everyday life. Or perhaps more accurately, the determinacies we think we observe in everyday life are illusory. In the final analysis the quantum paradigm applies across the board.