I suppose that the hypothesis of my paper, that the observed frequency of a photon is the frequency of classical electromagnetic fields resulting from the motion of its charged constituents, is perceived as self-contradictory by most physicists. If the photon is by definition the sole constituent of the electromagnetic field, as a couple of generations of physicists have been taught, then it is not possible that it can itself be a source of electromagnetic fields. Yet this hypothesis makes possible the derivation of a photon energy-frequency relationship that is arguably more correct than the standard relation proposed by Einstein, as it provides a resolution of the spin-orbit coupling anomaly more plausible than invocation of Thomas precession. (And recognizing that the Thomas precession as a kinematic effect cannot modify energy levels, it becomes possible to recognize it as accounting for the magnetic force and its effects, including previously-unrecognized magnetic binding of charged particles, that occurs irrespective of relative charge polarity.)

However perhaps the greatest explanatory utility of a photon with an electromagnetic structure, and whose interactions are describable through classical electrodynamics, is that it can allow electromagnetism and particularly the magnetic force to account for the so-called wave character of matter, without conflict with quantum electrodynamics and quantum field theory generally. A composite photon with an electromagnetic structure, interacting with charged matter in accordance with classical electrodynamics, obviates the need to postulate a non-classical wave character of matter, which has been shown by Bohm equivalent to postulating a new force that is of magnitude similar and sometimes equal to the Lorentz force. Why invent a new force when a known force can plausibly do the job?

## Saturday, December 30, 2017

## Monday, December 25, 2017

### Did I Make Algebra Errors?

I don't believe I did, but that's what the Foundations of Physics reviewer said about my photon paper (now posted on arxiv, with only narrative changes compared to the FOOP submittal here: Composite Photon Energy-Frequency Relation). But said reviewer didn't say anything specific about what the alleged errors are. I am extremely disappointed with FOOP for passing this off as an actual review. They apparently cannot be bothered to take a position on whether the composite massive photon does indeed have a derivable energy-frequency relationship that is consistent with the Planck law, when it is taken into account that two photons are needed to constitute a standing-wave mode as used by Planck in developing the first correct blackbody spectrum description.

I will post the review and a more detailed response later. There was also a second review that is demonstrably not even about my paper. I received the reviews and reject decision in early October. I wrote a letter to FOOP soon after asking for more specifics on what the alleged errors are, and pointing out that the other review was not even about my paper, but have not received any reply other than from the Springer front office that my letter would be forward appropriately.

There are several reasons I continue to have confidence that there are no substantive algebraic errors in my paper. The algebra in the paper is rather simple, and most of it is quite standard and published by others and in mainstream journals, as cited. It seems the reviewer has no familiarity with this literature and couldn't be bothered to look at the relevant references. In the use of the relativistic Doppler effect, the treatment is completely standard and in many textbooks. Algebra associated with Lorentz factors could be a bit surprising perhaps to those unfamiliar with it, but seriously I think it is reasonable to expect a professional physicist to be intimatey familiar with it. I tried to put essentially every step in the derivation embedded in either the narrative text or as numbered equations, to the point that I can check every step just on re-reading the paper. Rereading it post-review I could not see any errors to correct.

I also spent a lot of time going over the derivation, immediately after obtaining the result, hoping to eliminate the factor of one-half I'd obtained compared to the Planck-Einstein law (i.e. E = h nu). When I realized I could (probably) calculate an energy-frequency relation for the composite photon (as described in Don Lincoln's Scientific American article, "The Inner Life of Quarks"), I was wondering if I would get the Planck-Einstein relation, and thinking it would be a nice paper if I did. So I was irritated by the factor of half, which I viewed initially as an obstacle to publication and general acceptance, and looked long and hard for mistakes or a way to get rid of it. The fact that I couldn't get any other result however gave me a suspicion that there might be something more profound about what I'd found. Still, I set the problem aside for months because I was working on understanding the relativistic interaction between two zitter electrons.

My paper published by FOOP in 2016 only worked out the case of nonrelativistic motion of the interacting zitter particles (which are however intrinsically relativistic objects), and wasn't clear that this restriction was in effect, so I was trying hard to get the full result for submittal as a sequel. In working the interaction problem out more systematically, however, I soon realized that the factor of a half I'd obtained on the photon energy-frequency relationship compared to expectation could be related to the energy eigenvalues that would result from an equation I was able to obtain for the interaction of two zitter electrons. The equation is identical to a one-dimensional free-particle Schroedinger equation except that the frequency is doubled (its derivation is in the previous arxiv version to the link above, v7, which is a much longer version but has various issues that make it unsuitable for publication until they can be worked out or split off). This naturally leads to an expectation that photons with energy equatable with the difference between energy eigenvalues will be halved, in agreement with my result for the photon. It also provides an explanation for the spin-orbit coupling anomaly that doesn't require invoking the Thomas precession, which I have thought for a long time to be an obviously incorrect and poorly justified explanation. Halving of all energy levels (which is different than halving the Planck constant, a distinction missed by Reviewer 1) is a much more elegant resolution the spin-orbit coupling anomaly, which can coexist with the Dirac relativistic electron theory, and explain why the Thomas precession does not appear explicitly in the latter, which also does not suffer the anomaly of the semiclssical spin-orbit coupling analysis.

Having things appearing to come together in a more-compelling semiclassical resolution of the spin-orbit coupling anomaly was very exciting, but I was still intimidated by the problem of disagreement with Planck and Einstein, so for months I put off trying to reconcile my photon result with theirs. This was fine as long as I was making progress on the relativistic zitter particle interaction problem, but when that got bogged down I decided to attempt a reconcilation. I started with a close examination of how Planck derived successfully the first black-body spectrum. I wasn't expecting to find a mistake (and I didn't), but I thought maybe I could incorporate negative energy states, as these would not have been recognized as a possibility circa 1900, and by doubling the number of possible energy levels the factor of a half might arise in a convincing way. However I was unable to build an argument that worked around this idea. Then I started thinking about and looking at how Einstein incorporated the Planck result into his conjecture that light is corpuscular and realized there is a basic disconnect between a photon that carries nonzero momentum and the energy states in a cavity resonator that were studied by Planck. That is, the states studied by Planck (and Rayleigh and Jeans before him) were standing wave modes and as such cannot possess nonzero momentum. As is well known, standing waves can be understood as a superposition of two equal-amplitude oppositely traveling waves, which leads naturally to each standing wave mode of quantized energy consisting of an even number of oppositely-traveling photons within the cavity. This seems perfectly natural to me as a description of the cavity in thermodynamic equilibrium, where for every photon being radiated and absorbed in one direction, there is necessarily another in the opposite direction. Otherwise, a closed cavity oscillator in isolation will not maintain a constant temperature over its external surface, and will tend to accelerate spontaneously if placed in free space. So it seems to me that Einstein's adoption of the energy of Planck's quantum of energy for the cavity oscillator modes as the energy of a light corpuscle was not carefully thought out but rather a careless assumption.

Later I will post both of the FOOP reviews and try to comment more specifically on them, but for now I will just mention that I sent the slightly revised version of my paper, along with the FOOP submittal and reviews to EJTP (a peer-reviewed journal without publication fees) and they have acknowledged receiving it.

If anyone can point out any errors in the arxiv version I'd be grateful if they would post them here and/or email them to me. Unhappy but grateful nonetheless. Moderation is on so comments won't post right away but I promise I will approve any serious comment.

I will post the review and a more detailed response later. There was also a second review that is demonstrably not even about my paper. I received the reviews and reject decision in early October. I wrote a letter to FOOP soon after asking for more specifics on what the alleged errors are, and pointing out that the other review was not even about my paper, but have not received any reply other than from the Springer front office that my letter would be forward appropriately.

There are several reasons I continue to have confidence that there are no substantive algebraic errors in my paper. The algebra in the paper is rather simple, and most of it is quite standard and published by others and in mainstream journals, as cited. It seems the reviewer has no familiarity with this literature and couldn't be bothered to look at the relevant references. In the use of the relativistic Doppler effect, the treatment is completely standard and in many textbooks. Algebra associated with Lorentz factors could be a bit surprising perhaps to those unfamiliar with it, but seriously I think it is reasonable to expect a professional physicist to be intimatey familiar with it. I tried to put essentially every step in the derivation embedded in either the narrative text or as numbered equations, to the point that I can check every step just on re-reading the paper. Rereading it post-review I could not see any errors to correct.

I also spent a lot of time going over the derivation, immediately after obtaining the result, hoping to eliminate the factor of one-half I'd obtained compared to the Planck-Einstein law (i.e. E = h nu). When I realized I could (probably) calculate an energy-frequency relation for the composite photon (as described in Don Lincoln's Scientific American article, "The Inner Life of Quarks"), I was wondering if I would get the Planck-Einstein relation, and thinking it would be a nice paper if I did. So I was irritated by the factor of half, which I viewed initially as an obstacle to publication and general acceptance, and looked long and hard for mistakes or a way to get rid of it. The fact that I couldn't get any other result however gave me a suspicion that there might be something more profound about what I'd found. Still, I set the problem aside for months because I was working on understanding the relativistic interaction between two zitter electrons.

My paper published by FOOP in 2016 only worked out the case of nonrelativistic motion of the interacting zitter particles (which are however intrinsically relativistic objects), and wasn't clear that this restriction was in effect, so I was trying hard to get the full result for submittal as a sequel. In working the interaction problem out more systematically, however, I soon realized that the factor of a half I'd obtained on the photon energy-frequency relationship compared to expectation could be related to the energy eigenvalues that would result from an equation I was able to obtain for the interaction of two zitter electrons. The equation is identical to a one-dimensional free-particle Schroedinger equation except that the frequency is doubled (its derivation is in the previous arxiv version to the link above, v7, which is a much longer version but has various issues that make it unsuitable for publication until they can be worked out or split off). This naturally leads to an expectation that photons with energy equatable with the difference between energy eigenvalues will be halved, in agreement with my result for the photon. It also provides an explanation for the spin-orbit coupling anomaly that doesn't require invoking the Thomas precession, which I have thought for a long time to be an obviously incorrect and poorly justified explanation. Halving of all energy levels (which is different than halving the Planck constant, a distinction missed by Reviewer 1) is a much more elegant resolution the spin-orbit coupling anomaly, which can coexist with the Dirac relativistic electron theory, and explain why the Thomas precession does not appear explicitly in the latter, which also does not suffer the anomaly of the semiclssical spin-orbit coupling analysis.

Having things appearing to come together in a more-compelling semiclassical resolution of the spin-orbit coupling anomaly was very exciting, but I was still intimidated by the problem of disagreement with Planck and Einstein, so for months I put off trying to reconcile my photon result with theirs. This was fine as long as I was making progress on the relativistic zitter particle interaction problem, but when that got bogged down I decided to attempt a reconcilation. I started with a close examination of how Planck derived successfully the first black-body spectrum. I wasn't expecting to find a mistake (and I didn't), but I thought maybe I could incorporate negative energy states, as these would not have been recognized as a possibility circa 1900, and by doubling the number of possible energy levels the factor of a half might arise in a convincing way. However I was unable to build an argument that worked around this idea. Then I started thinking about and looking at how Einstein incorporated the Planck result into his conjecture that light is corpuscular and realized there is a basic disconnect between a photon that carries nonzero momentum and the energy states in a cavity resonator that were studied by Planck. That is, the states studied by Planck (and Rayleigh and Jeans before him) were standing wave modes and as such cannot possess nonzero momentum. As is well known, standing waves can be understood as a superposition of two equal-amplitude oppositely traveling waves, which leads naturally to each standing wave mode of quantized energy consisting of an even number of oppositely-traveling photons within the cavity. This seems perfectly natural to me as a description of the cavity in thermodynamic equilibrium, where for every photon being radiated and absorbed in one direction, there is necessarily another in the opposite direction. Otherwise, a closed cavity oscillator in isolation will not maintain a constant temperature over its external surface, and will tend to accelerate spontaneously if placed in free space. So it seems to me that Einstein's adoption of the energy of Planck's quantum of energy for the cavity oscillator modes as the energy of a light corpuscle was not carefully thought out but rather a careless assumption.

Later I will post both of the FOOP reviews and try to comment more specifically on them, but for now I will just mention that I sent the slightly revised version of my paper, along with the FOOP submittal and reviews to EJTP (a peer-reviewed journal without publication fees) and they have acknowledged receiving it.

If anyone can point out any errors in the arxiv version I'd be grateful if they would post them here and/or email them to me. Unhappy but grateful nonetheless. Moderation is on so comments won't post right away but I promise I will approve any serious comment.

## Monday, October 2, 2017

### Status Update on My Photon Paper

My paper on the energy-frequency relation of a composite photon was sent out for review by Foundations of Physics, and their website reports they are complete as of September 14, but I haven't yet seen them or been informed of a decision.

I want to mention that essentially all of the content of the submitted paper is in the larger arxiv paper here: https://arxiv.org/abs/1609.04446

The longer paper I think better motivates the content of the submitted photon paper by putting it in a larger context. But it's long and still has some problems so I didn't think it wise to submit in its entirety.

I want to mention that essentially all of the content of the submitted paper is in the larger arxiv paper here: https://arxiv.org/abs/1609.04446

The longer paper I think better motivates the content of the submitted photon paper by putting it in a larger context. But it's long and still has some problems so I didn't think it wise to submit in its entirety.

## Saturday, July 15, 2017

### My photon paper is publicly viewable

I extracted from my longer paper on arxiv all the parts about the massive composite photon, and how it leads to a correction to the Planck-Einstein relation and explains the spin-orbit coupling anomaly without invoking Thomas precession, and put it together into a short paper. I submitted the paper to Foundations of Physics a couple of days ago. Now I have posted it on Researchgate: Composite Photon Energy-Frequency Relation.

## Sunday, June 25, 2017

### One Photon Does Not Make a Standing Wave

I have finally figured out how to reconcile the composite massive photon model, that gives E equals h nu over 2, with the Planck-Einstein law, E equals h nu. It's a very simple reconcilation that requires no modification of Planck's blackbody spectrum, or of the value of the Planck constant h.

When Einstein realized that quantization of light explained the photoelectric effect, he assumed that the light quanta carried energy in accordance with Planck's assumption that the standing wave electromagnetic modes inside a cavity had energy quantized as integer multiples of h nu, where h is as found by Planck (and to within a few percent by Planck himself). However, a photon must necessarily carry momentum E/c, where c is the speed of light, while electromagnetic standing waves in a cavity and at therrmodynamic equilibrium cannot possess net linear momentum. Therefore the quantized energy value of a standing wave mode in multiples of h nu and the number of photons in the cavity at frequency nu cannot be directly equated. Rather, since a standing wave can be regarded as a superposition of two oppositely-traveling waves of equal amplitude, the expected number of photons is twice the number of energy multiples of h nu. This is in agreement, to within all measurements of the speed of light to date, with the frequency-to-energy relationship for the composite photon, which is easily derivable based on the relativistic Doppler effect.

After having thought about the possible ways that a photon of half the currently-accepted energy can be reconciled with established physics, if at all, this seems best to me. In particular, it does not require revision of the value of h, which would lead to changes in the size of atoms (as far as I can see), in addition to many other problems. On the other hand, retaining the Planck blackbody spectrum while reducing the photon energy by half, and retaining that the angular momentum of a photon is h-bar=h/(2 pi), requires only an energy recalibration. To begin with energy levels on emission spectra can be divided by two. This leads immediately to a resolution of the spin-orbit coupling anomaly that does not need to invoke the Thomas precession. It is also straightforward to modify the Schroedinger equation to be in agreement with the photon carrying half the energy.

A new version of my paper is slated to appear at midnight GMT tonight, that will reflect my new point of view. It will be version 7 here: https://arxiv.org/abs/1609.04446

When Einstein realized that quantization of light explained the photoelectric effect, he assumed that the light quanta carried energy in accordance with Planck's assumption that the standing wave electromagnetic modes inside a cavity had energy quantized as integer multiples of h nu, where h is as found by Planck (and to within a few percent by Planck himself). However, a photon must necessarily carry momentum E/c, where c is the speed of light, while electromagnetic standing waves in a cavity and at therrmodynamic equilibrium cannot possess net linear momentum. Therefore the quantized energy value of a standing wave mode in multiples of h nu and the number of photons in the cavity at frequency nu cannot be directly equated. Rather, since a standing wave can be regarded as a superposition of two oppositely-traveling waves of equal amplitude, the expected number of photons is twice the number of energy multiples of h nu. This is in agreement, to within all measurements of the speed of light to date, with the frequency-to-energy relationship for the composite photon, which is easily derivable based on the relativistic Doppler effect.

After having thought about the possible ways that a photon of half the currently-accepted energy can be reconciled with established physics, if at all, this seems best to me. In particular, it does not require revision of the value of h, which would lead to changes in the size of atoms (as far as I can see), in addition to many other problems. On the other hand, retaining the Planck blackbody spectrum while reducing the photon energy by half, and retaining that the angular momentum of a photon is h-bar=h/(2 pi), requires only an energy recalibration. To begin with energy levels on emission spectra can be divided by two. This leads immediately to a resolution of the spin-orbit coupling anomaly that does not need to invoke the Thomas precession. It is also straightforward to modify the Schroedinger equation to be in agreement with the photon carrying half the energy.

A new version of my paper is slated to appear at midnight GMT tonight, that will reflect my new point of view. It will be version 7 here: https://arxiv.org/abs/1609.04446

## Tuesday, May 23, 2017

### A possible alternative explanation for the spin-orbit coupling anomaly

I've been considering my hypothesis that Planck's constant is really twice the accepted value, based on that the photon energy is related to its frequency by half the accepted amount, for the composite photon, as I proposed two posts ago. The hypothesis fails in an obvious way. Since it implies the electron g-factor is actually unity, it is inconsistent with the Zeeman effect, where it is essential that the electron g-factor be (close to) two, in order to agree with observation.

I now have an alternative hypothesis about the photon energy being half the accepted amount in relation to its frequency. That hypothesis is that it explains the spin-orbit coupling anomaly as well or better than the Thomas precession does, while also not otherwise differing noticeably from observation.

If the photon energy for a given atomic transition is half that currently assigned (that is, if the current energy assigned is twice the correct value), then the measure of the spin-orbit coupling energy from emission spectra will appear to be doubled as well. So, correcting the transition energy to be half the accepted value will recover the spin-orbit coupling energy consistent with the original prediction. There will be no need to apply the Thomas factor of one-half in order to find agreement with observation.

As I think I have stated previously, and various people have been pointing out for many decades, it makes no sense that the Thomas precession, a purely kinematical effect, can alter the interaction energy.

Halving the energy per photon has another consequence of halving the expected magnitude of the Zeeman effect generally. I'm not sure if this would be noticeable or not.

I now have an alternative hypothesis about the photon energy being half the accepted amount in relation to its frequency. That hypothesis is that it explains the spin-orbit coupling anomaly as well or better than the Thomas precession does, while also not otherwise differing noticeably from observation.

If the photon energy for a given atomic transition is half that currently assigned (that is, if the current energy assigned is twice the correct value), then the measure of the spin-orbit coupling energy from emission spectra will appear to be doubled as well. So, correcting the transition energy to be half the accepted value will recover the spin-orbit coupling energy consistent with the original prediction. There will be no need to apply the Thomas factor of one-half in order to find agreement with observation.

As I think I have stated previously, and various people have been pointing out for many decades, it makes no sense that the Thomas precession, a purely kinematical effect, can alter the interaction energy.

Halving the energy per photon has another consequence of halving the expected magnitude of the Zeeman effect generally. I'm not sure if this would be noticeable or not.

## Saturday, March 11, 2017

### More extensions to my paper on ArXiv

This version (v5) obtains that the modulating factor on the inverse-square radial magnetic force between equal mass and parallel-spin zitter particles satisfies the three-dimensional time-dependent Schroedinger equation, except that hbar multiplying the partial derivative with respect to time is replaced by hbar/2. This equation implies an angular momentum spectrum in terms of integer multiples of hbar, but an energy spectrum in terms of integer multiples of hbar/2. This is consistent both with the energy per photon as derived in the same paper for a composite photon to be h/2 times the photon frequency, and so that the accepted value of Planck's constant (h) is only half the correct value. This also implies that the electron g-factor is approximately unity, rather than approximately 2 as currently thought.

Spin-Spin Interaction as Quantum Phase Measurement and the Mechanism of Atom Formation

Next I will go over everything in it again and try to shorten it, and then submit it to a journal. Since Foundations of Physics published my earlier paper from which this one derives, I plan to submit there.

There is much more I would like to do on it, in particular to try to derive the same result using Hamilton-Jacobi theory, but I have no idea if I can succeed at that or how long it might take, so I think it is a good enough time to submit. I hope at least they will review it.

Spin-Spin Interaction as Quantum Phase Measurement and the Mechanism of Atom Formation

Next I will go over everything in it again and try to shorten it, and then submit it to a journal. Since Foundations of Physics published my earlier paper from which this one derives, I plan to submit there.

There is much more I would like to do on it, in particular to try to derive the same result using Hamilton-Jacobi theory, but I have no idea if I can succeed at that or how long it might take, so I think it is a good enough time to submit. I hope at least they will review it.

## Thursday, January 12, 2017

### Why the electron g-factor is 2

It's because the generally accepted value of Planck's constant is only half the correct value.

The quantity h-bar in the Schroedinger equation is actually h-bar/2. That's because it represents the spin of the electron, not of the photon, as implicitly assumed by de Broglie and carried forward by Schroedinger, and everyone else.

This fact has been overlooked because it was not recognized that the photon frequency is related to its energy not as E = h nu, but as E = h nu /(1 + v_l/c), where c is the limiting velocity of relativity theory, and v_l is the velocity of the photon, which is only very slightly less than c.

Although I derived that the energy of a composite photon is half the accepted value several months ago, and posted it about here previously, I didn't realize the implications until a few days ago, when I realized that the modulating factor on the Coulomb-like magnetic force between zitter particles satisfies the Schroedinger equation for a free particle in one spatial dimension. Except, the hbar must be replaced by hbar/2. This makes the energy difference between atomic energy levels half as large as previously thought, consistent with the photon energy being half the accepted value. However, simply replacing the hbar in the SE by hbar/2 leads obviously also to an angular momentum spectrum in multiples of hbar/2 rather than hbar. This cannot be, and the photon energy being hbar/2 can only be correct if, in addition to hbar being twice the currently-accepted value, hbar remains the quantum unit of angular momentum. So, the form of the SE has to change from its current form.

The modulation factor that I derived is equivalent to the Schroedinger wavefunction in one dimension, but not in three dimensions. There is a preferred dimension defined by the null four-displacement between the zitter particles. It is essentially the relativistic Doppler effect along the radial direction that gives rise to the modulating factor that can be equated with the de Broglie wave if hbar is replaced by hbar/2. (De Broglie was reasoning by analogy with the photon without knowing of the intrinsic angular momentum difference of the electron. Had he known of the difference, he would have expected the hbar/2.) In the cross-radial direction, though, the modulation goes over to a de Broglie wavelength (rather than half it for radial motion), as I showed in my FOOP paper. So it seems plausible that the Schroedinger-like equation based on the magnetic force modulation could lead to an energy spectrum in terms of multiples of hbar/2 but an angular momentum spectrum that agrees with observation, but that is work to do at the moment.

Sorry there are some typos I'll fix in the next revision. Current is v4: https://arxiv.org/abs/1609.04446

The quantity h-bar in the Schroedinger equation is actually h-bar/2. That's because it represents the spin of the electron, not of the photon, as implicitly assumed by de Broglie and carried forward by Schroedinger, and everyone else.

This fact has been overlooked because it was not recognized that the photon frequency is related to its energy not as E = h nu, but as E = h nu /(1 + v_l/c), where c is the limiting velocity of relativity theory, and v_l is the velocity of the photon, which is only very slightly less than c.

Although I derived that the energy of a composite photon is half the accepted value several months ago, and posted it about here previously, I didn't realize the implications until a few days ago, when I realized that the modulating factor on the Coulomb-like magnetic force between zitter particles satisfies the Schroedinger equation for a free particle in one spatial dimension. Except, the hbar must be replaced by hbar/2. This makes the energy difference between atomic energy levels half as large as previously thought, consistent with the photon energy being half the accepted value. However, simply replacing the hbar in the SE by hbar/2 leads obviously also to an angular momentum spectrum in multiples of hbar/2 rather than hbar. This cannot be, and the photon energy being hbar/2 can only be correct if, in addition to hbar being twice the currently-accepted value, hbar remains the quantum unit of angular momentum. So, the form of the SE has to change from its current form.

The modulation factor that I derived is equivalent to the Schroedinger wavefunction in one dimension, but not in three dimensions. There is a preferred dimension defined by the null four-displacement between the zitter particles. It is essentially the relativistic Doppler effect along the radial direction that gives rise to the modulating factor that can be equated with the de Broglie wave if hbar is replaced by hbar/2. (De Broglie was reasoning by analogy with the photon without knowing of the intrinsic angular momentum difference of the electron. Had he known of the difference, he would have expected the hbar/2.) In the cross-radial direction, though, the modulation goes over to a de Broglie wavelength (rather than half it for radial motion), as I showed in my FOOP paper. So it seems plausible that the Schroedinger-like equation based on the magnetic force modulation could lead to an energy spectrum in terms of multiples of hbar/2 but an angular momentum spectrum that agrees with observation, but that is work to do at the moment.

Sorry there are some typos I'll fix in the next revision. Current is v4: https://arxiv.org/abs/1609.04446

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