You mean the intrinsic spin of the electron? I would not try to visualize it. The history is that Pauli tried a mechanical model, which required too much energy. After which Uhlenbeck and Goudsmit tried to withdraw their paper.

The most elegant way to describe electron spin "classically" is in terms of Grassman "numbers", which are not easy to visualize.

What do you mean by saying that each ns state in the hydrogen atom only consists of one state? That the two spin states of the electron combine with the two spin states of the proton to form a singlet an triplet? Or something else? 

Dear Xavier and Kåre ,

This question is very interesting. I would like to contribute some modest pieces of information about this issue in which I am no specialist.

As we know, the intrinsic spin of the electron is proportional with the magnetic momentum of the electron, though, the magnetic momentum is not due to a rotation of the electron around itself. From this situation we can infer that inside the electron there exists some dynamics, more complicated than just a rotation of a charge around itself. Further, this dynamics leads us to think that the electron has some internal composition, or some internal structure.

There exist theoretical researches that indicate that the electron has an internal composition. However, they also show that for breaking the electron into pieces there have to be invested huge energies.

As we  know, when energies are needed to break down an object, that means that the parts of the object are held together by forces. So, we invest energy for overcoming those forces. 

Bottom line, this question invites another one: could it be that inside the electron there exists a new type of force besides those known to us until now, and maybe of an even shorter range?

With best regards,

Sofia

Here you speak about the orbital angular momentum.  It is 0 (in unit of hbar) for the ns shell and has only 1 projection on an arbitrary axis.  It is 1 for the np shell with 3 different projections, and 2 for the nd one with 5 projections.  The orbital angular momentum is always an integer.

The spin, or intrinsic angular momentum, appears only in the Dirac equation.  Its magnitude is 1/2 and has 2 projections, whatever the motion of the electron.  Loosely speaking, it is analogous to the angular momentum of a top.

The orbital anglular momentum is exhibited by the normal Zeeman effect, while the spin is by the anomalous one.  The latter is called so because it wasn't predicted by the Schrödinger equation, and is much weaker.

Dear All,

 Dirac in 1928 offered a new equation which has all the levels of energy i.e. doublets levels in addition to those already obtained. One important aspect is that the classical momentum k of Sommerfeld take the following value k = l or k = - (l +1) for l ≥ 0, but the momentum is not the angular momentum. The study of these solutions, assuming the speed of light toward infinity, shows that they tend to those of Schrödinger, with the following conditions for the values for k;  k = l and k = - (l + 1) with k different of zero, and l ≥ 0. The “s” states correspond to the case k = - (l + 1) with l=0, so to k = -1.

This first established that the classical approach of the Schrödinger solutions is not sufficient.

The word intrinsic means belonging to the real nature of the thing, but the dictionary adds not dependent of external circumstance. This second point is in contradiction with the notion of motion or rotation. Does really the motion here the spin rotation can be a property without connection with the proton, or if you prefer the orbital rotation can really be independent of the spin rotation?

In fact in the classical approach the spin is supposed to be added or subtracted to the orbital angular momentum it is a kind of connection with the orbital motion. But it is not that the Dirac solutions of its equation show. The addition of one momentum to the “ns” states gives the all the quantum states and nevertheless the half integer appear in the angular momentum. Furthermore the number of states are 2l for k = l and 2(l + 1) for k = - ( l + 1).

In my opinion the first quantum h giving the “ns” states induce the quantification of all the space that is the angular momentum but also the direction perpendicular to the direction of the plane of projection of the angular momentum. In other words the direction of the axis of orbital rotation is also quantified. This implies that the first quantum h is divided in two equal parts, so the half-integer number.

To answer to the question: How do you visualize the two kinds of spin of an electron?

I suppose that the spin and the orbital rotations are necessary simultaneous with just one possibility.

Now Sofia you write: Bottom line, this question invites another one: could it be that inside the electron there exists a new type of force besides those known to us until now, and may be of an even shorter range? From my point of view this remark concerns the interaction it is another subject. I will try later to have something else to say.

 Best regards.

Xavier

I conclude it was something else.

But, apart from your wish for an old-quantum-theory understanding of the hydrogen atom, do you think there is anything wrong with the solution given in many textbooks? It is my impression that the Dirac equation reproduces the spectrum (and many other properties) very well, with known ways to compute QED corrections. Or is it just a wish to obtain a semi-classical understanding of half-integer spin?

Dear Kåre,

Bohr in is paper in 1913, to propose an interpretation of his results, wrote page 15: “If we assume that the orbit of the electron in a stationary states is circular, the result of the calculation on page 5 can be expressed in a simple condition: that the angular momentum of the electron round the nucleus in a stationary state of the system is equal to an entire multiple of an universal value, independent of the charge of the nucleus.”

As a result the Bohr magneton with the Planck constant are still supposed to be at the foundations of the quantum state. Since my paper “The quantum state and the doublets” Ann. Fondation Louis de Broglie, 2000, 25, 1-25, I suppose that there is no reason for which the nature would forget the perpendicular direction to the angular momentum and quantifies it also.

Now is there something wrong?

Considering the Schrödinger equation it is evident from the equation of Dirac in his original paper that the angular momentum cannot be zero. Proc. Roy. Soc. A117, page 610-624, see page 623.

Now,  probably few authors know the point that I raise. But you must know that I mainly read the two first papers of Dirac about its equation, a little is book fourth edition, and and I work out the Dirac’s solutions of its equation with  “L’élecron magnétique” by Louis de Broglie. In magnetism very few people know these solutions, especially in the interpretation of the experimental magnetism.

So I suppose that textbook, too much involve in the wave approach, miss the point.

In fact after the paper of Einstein on the special relativity, people does not easily forgotten the notion of ether and wave. So the work of Louis de Broglie is still used to explain the propagation of the light. But as I explain elsewhere we have to choice between the Continuous and the discreteness. The « Wave and the Quantum State », Annales de la Fondation Louis de Broglie 40, 211-221, 2015.

Finally Dirac’s equation describe the variations of the mechanical action along the trajectory, something not easy to visualize, it is perhaps the reason of the shadow on the quantum mechanics.

> Now is there something wrong?

The old quantum theory was an important step towards the quantum theory of matter, but seems to me today like a dead end -- how should one approach non-integrable systems, like the Helium atom, this way? Of course, one can find traces of the old quantum theory in the new one, in approaches like the WKB approximation, and at other viewpoints appealing to physical intuition.

A (semi-)classical description of objects with half-integer spin (as in the work by Jackiw and Rebbi) is interesting, but I kind of doubt that it can be convincingly found in the Dirac equation for the hydrogen atom.

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.36.1116

Dear All,

It is wrong to make assumption before to discuss the notion of space.

Einstein in 1905 with his study of relativity showed that there is no absolute space and that physical laws should be independent of the place of observation, which made a giant leap in physics. As a result it did not come to anybody the idea that there could be a lack in this analysis. Yet the difficulties encountered by the corpuscular approach of quantum mechanics are well related to the notion of space. Indeed there is no absolute space and one has to question how the space is built at the scale of the atom? Up to now electromagnetic field has allowed us to write the motion of the electron around the nucleus. But Einstein has also taught us that the mass is equivalent to an amount of energy by establishing the relationship:

This energy is huge so that it suggests that the electromagnetic field can be described by a discontinuous flow of grains of matter extremely small, constituting the mass, between the proton and the electron. More precisely through two fluxes: one from the proton to the electron the other from the electron to proton, these two fluxes not following everywhere the same route in such a way that the resulting action creates the movement. The energy thus involved is that of the quantum state of the electron, it is very small compared to the mass of each of the two particles. This approach allows to be in agreement with the conclusion of Einstein. When the electron is at the infinite it has received the energy that linked it to the proton, it has his total inert mass and there is no more wave function to consider.

Now we must be careful in using mathematic. The language of mathematics is remarkable and I always liked it; I remember Professor Laurent Schwartz teaching us symbolic algebra where operators can be manipulated as numbers that solve differential equations, I was full of admiration and I still remain so. But despite the elegance of the method, the difficulties of continuous and infinite remain and I think that the nature is discontinuous.

I've always fascinated by the concept of "electron spin", and wanted to find at least an mechanical analog of it. Recently, non-linear elastic theory is a kind of ``rediscovered" burgeoning field, in which the the classical Newtonian particles are replaced with particles with "intrinsic spins". We all know that spin tops act differently from no-spinning objects, such as climbing a slope, or moving in the orthogonal direction to a push. Is is possible to try similar things to the ``tiny spinning tops" of electrons?

I never whole-heatedly embrace the quantum theory, and to be honest, feel that certain developments in the relevant fields have gone to far to absurdity. I'd rather looking in a direcition of an alternative theory such as stochastic mechanics.

Dear all,

The spin notion can be approached as follows: consider the two particles proton and electron. Consider the proton with its center of mass at the origin of an orthogonal reference frame. Is it possible that the electron does not has a rotation call spin in addition to its orbital motion? I think no, this will mean a strong connection between the two particles. Now according to Einstein “physical laws should be independent of the place of observation”, the exchange of mass in the form of very small grains, seems to be the alone way to have the same law for the interaction if we consider the motion from the proton or from the electron.  On the same way of approach it must be the mass which induces the orbital and spin motions.

Now with just the exchanges of mass there is just one possibility of spin rotation QED.  

You will find a discussion of the Stern and Gerlach experiment in “Quantum and Periodicity” and also in “L’état quantique, le magnétisme et la rotation” that you can consult on my ReaserchGate site.

Dear Fushan Zhou,

would you be so kind to mention some examples where you find that absurdity concerning quantum mechanics? I'd appreciate that in order to get an impession of the meaning of what you're trying to say.

With the electron spin, though, we know that a simple analog (rotating charged something of finite size) will not work, without giving up special relativity -- at least this is my (maybe naive) understanding.

Dear Kåre and All

You asked me: Now is there something wrong? I thing that Dirac was wrong about the electric moment which he found. Indeed I wish in this spot to tackle a point often do not known, Dirac in is paper giving all the quantum states, found that the electron behave as it has a magnetic moment but also an electric moment. Dirac does not expect that the electric moment being a pure imaginary has any physical meaning. In fact we can interpret that the imaginary character is there to separate it from the magnetic moment a classical way to work simultaneously independent quantities. 

This electric moment is thus parallel to the magnetic moment and establishes the quantification of the axe of rotation.

So this magnetic moment is probably the property observed in compound as in Perovskites [ABX3] having an electric moment? In fact there are not so much ferroelectric compounds as magnetic moment but this must probably the result of the strong interaction of electric dipoles.

The spin is intrinsic in the meaning that it doesn't depend on the motion of the particle.  An electron at rest still have a spin 1/2.  In contrast, any particle has an orbital momentum that depends on the origin of the referential frame. It is discrete only for closed orbits.

Dear Claude,

I have to disagree to some extent:

take some hydrogen orbital (disregarding spin and spin orbit coupling for the moment) with l unequal zero and  ml also unequal to zero. In the reference frame centered at the nucles (infinite mass assumed for simplicity) this state is an eigenstate of both L2 and Lz with eigenvalues (setting Plancks constant to unity) l(l+1) and ml, respectively.

If you choose a different origin, then this is no longer true. Nevertheless, the expectation values of these operators in the displaced setting are the same as the eigenvalues in the centered reference frame.

Therefore, when you make a lattice of such hydrogen "atoms" (in this state) and construct Bloch states as the eigenstates of the lattice from these local orbitals, then even those Bloch states share the same expectation values of L2 and Lz.

Dear Xavier,

a note on ferroelectrics first: ferroelectric materils necessarily are insulators. Their finite spontaneous dielectric polarization is connected with atomic displacements of ions - away from symmetric positions. This simply means that they develop a potential of "mexican-hat" type, where the minimum is not in the center of the "cage" given by the surrounding atoms. (This is pretty well-known and can be confirmed, e.g. with Xray diffraction as a function of temperature and other experiments.)

And then as to the "electric moment" you are contemplating: do you mean the electron does ave a finite electric dipole moment? If that was the case, then the interaction would not be of Coulombic shape, not possess spherical symmetry. Is this seriously what you think is true?

Kai, at a point far from a hydrogen atom, the average angular momentum is zero.  That doesn't mean there is only one eigenstate for the electron.  If the electron is in an angular momentum eigenstate with respect to the proton, the measurement according to the far angular momentum operator may give different values of the angular momentum, but weighted with the probabilities they give zero.  In other words, the angular momentum operators for different origins don't commute.

One more question, relating to the OP:

When you say: ''Now according to Einstein “physical laws should be independent of the place of observation”, the exchange of mass in the form of very small grains, seems to be the alone way to have the same law for the interaction if we consider the motion from the proton or from the electron.''

I thought this would not be the case for accelerated reference frame. When you fix the proton at the origin (equivalent to attributing infinite mass to it) then this is one thing, but in case of the electron (if you think of it as a tiny object orbiting around the nucleus) this is certainly not valid.

Am I wrong here?

Kai,

Yes, I can give you examples. In QM, the moment when you take an experiment the wave function suddenly "collapse" to one of the eigen-functions of some operators. Can you explain how did this happen, or it just happened for no reason?

In QM theory, it is often said our minds can interfere with experimental results. If this is so, QM should be seriously considered a branch of psychology. Could you give more details on this?

A lot of "philosophical" or "theoretical"  theories thrived on QM, which, in my opinion, have already crossed the line between science and pseudo-science. So How do you judge a theory like QM if it leaves the door open for pseudo-science to creep in?

Fushan,

let me give some thoughts in different parts, to keep posts limited in length.

Pseudo-science is by far not limited to QM! I think, for example, about all those folks convinced about the existence of energy harvesting machines as the Bessler wheel an the likes. Pure classical mechanics here. And I would tend to think it is more telling about people than any scientific theory.

Whenever we deal with things that are outside our common experience, there is a risk that we might meet a situation where what we find might defy the way we think. This has fundamentally been the case with physical discoveries of the last 150 years. We were forced to somehow accept that mechanical velocities cannot add to more than the speed of light. This nonlinearity, embodied in the Lorentz transformation reveals something about spacetime which escapes us in normal everyday life. So, "how can this happen"? Is this explainable? What would be the fundamental premises on which we would base our judgement of the answer to this question? Or do we have to somehow change our mindset and accept that this seems to be the way things aroud us work? Different people might come to different conclusions. Is SRT to blame for this? If you ask me, the answer is "no". And I recognize for example that todays GPS systems only work with the actual accuracy because the rules of SRT are built in to the machinery (interpretation of experiments, if you like).

@ Fushan, ctd.

I view the "collapse of the wave function" as our way in QM to deal with conditional probabilities. If you throw two dices, and know nothing, then the probability for getting the sequence "3,5" is 1/36, just like the probability for the sequence "2,5" (assuming the dices to be fair :-). But if you somehow came to know, that the first dice gave a "2", then you know that the conditional probability for getting "3,5" is zero in this experiment, the one for "2,5" is 1/6. This does not invalidate the value 1/36 obtained in the first place for both choices.

This is maybe one way of acknowledging the fact that in the "microscopic world", doing a measurement has an impact on the object under observation. This is different from our daily experience that we can be "distant observers". This latter view is embodied in our teaching of classical physics, when we talk about forces on test masses or test charges probing gravitational or electric fields but being so small that they don't disturb the field being probed. These hypotheses are truly idealizations. These idealizations work fine in many cases but obviously have limitations.

In classical physics, this idealization allows us to make a theory about the object. In QM, the "measurement process" has to be built in. The prescription is to use Hermitian operators for this purpose etc.etc. and all the stuff you know. Can we judge "how that works"? What kind of process is involved in forming that judgement? How do we verify our judgement? How to verify whether my thinking is correct? (What is "thinking"....) I believe chances are high that we continue to be challenged...

Now in physics, the idea would be: let me take "the theory" serious and design experiments for which "crazy" results would be predicted (entanglement, delayed choice, interferometry with atoms and molecules etc.) One group being famous for having done a lot in this respect is the one of Zeilinger in Austria. As far as I know, all these results are in agreement with QM predictions. So, "nature" actually defies our thinking. With the rules of QM we seem to have found a pretty impressive way of describing the outcome of experiments. For some, this suffices to be satisfied, for others it doesn't. Is QM as a theory to be blamed for that or to be rejected right away for the discomfort it produces to my thinking? (And just to make this clear: I feel challenged at times by QM....)

@ Fushan, ctd.

Concerning the interference with the mind: of course the question comes up, "what is the thing we call "quantum mechanical state" actually describing (and "state" would be the general object, "wave function" is an instance of "state" if I choose a specific way of doing QM)? And if I propose formulations like "the state (wave function) contains everything I could ever know about the object", then does that explicitely involve myself in the contents of "state" ?

I now that experiments on adioactive decay were made, were it was attempted to find out whether it made a difference of whether some person took notice of the measurements being made or not. As far as I know the outcome was negative.

And when you say: "In QM theory, it is often said our minds can interfere with experimental results", who sais this, how is this (attempted to be) verified, and what is "often"?

(This put aside one can, of course, take aspects of QM (especially many body QM) as an invitation to philosophical ways of thinking. Schrödinger himself offers a good example, in that (afaik) he seems to have adhered to the idea that the inseparability inherent to QM somehow leads to a world in which 'everything' is somehow interlinked.)

Dear Claude Pierre Massé,

you propose: "at a point far from a hydrogen atom, the average angular momentum is zero."

Have you done the calculation? I'd really like to see the derivation of that result.

In classical mechanics at least your statement is not correct. Take a point object (with nonzero mass :-) on a periodic, circular trajectory about the origin. In that reference frame, the angular momentum (vector product of position vector and momentum vector) is a constant of motion.

If you choose a different reference frame, this is, in general, no longer the case. The angular momentum will now be a periodic in time. However, calculating the average over the period of revolution, you obtain the same result as the (constant) value in the origin. [This is a calculation I have done.]

@ Claude Pierre Massé, ctd.

Your statement "In other words, the angular momentum operators for different origins don't commute" means to me that these two operators do not have a common set of eigenstates. I immediately subscribe to that for two angular momentum operators (L1 and L2, say,) with different origin.

However this does not automatically imply that if an electron is in some eigenstate |n,l,ml> of L12 and L1,z (i.e.  L1,z|n,l,ml> = ml |n,l,ml> etc.[hbar=1 assumed]) that e.g. the expectation value is zero.

Kai, you are right for an orbiting motion. Yet, for a linear motion, the angular momentum depends on the origin.  Anyway, that was not my point. The spin doesn't need a reference to ordinary space to be defined, it is a function of internal variables that have a value at every point of space-time. That's why it is called intrinsic angular momentum. Precisely, the spin operator lives in the Dirac algebra, while the orbital angular momentum one is built with space derivatives.

Kai, good thing you are attacking the problem, not something else. You know people joke about whether a tree falls in a forrest makes sense if nobody is present, or whether the moon really exists if there is nobody to see it. We could have never let QM go into this type troubled water. This is beyond physics.

What is the spin of an electron? All textbooks say something like "the intrinsic angular momentum of an electron". Beyong that, I  don't believe there is anybody that can explain why and how an electrons possess such kind of thing.

Here,I just suggest that we look for the answer in another direction -- classical mechanics. Yes! Classical mechanics, Specifically, generalized elasticity of continua with microstructures, which sadly for various reasons, is virtually an unknown field to theoretically physicists.

Dear Fushan,

is "having spin" more fundamentally strange to you than "having mass" or "having charge"? (if yes, can you say why this is so?)

What makes you think that the mechanical viewpoint you wish to take should make spin less mysterious? There are several things that would be required to come out of this: half inter value for electrons (& neutrons & protons) gyromagnetic factors (magnetic moments associated with spin) and the spin statistics theorem (distinction of bosons and fermions via spin, with their known properties). It is beyond my imagination how an elastic model within classical mechanics would produce all this. Also, what evidence is there that the electron is a mechanically elastic object? How would that be desribed?

Dear Fushan,

in your first post, you mentioned "stochastic mechanics" as well. May I suggest you to study (the derivation of) the theorem due to van Leeuwen and (independently) N. Bohr? From my perspective, it is quite telling as far as the incompatibility of magnetism and classical physics is concerned.

Put into simple words: if magnetic effects (such as a magnetic moment acting as an angular momentum) were due to a top-like motion of a charged something, then why doesn't it radiate e.m. radiation? (And if it did, where would the energy be taken from and what is the evolution of the object then etc. etc...)

For the orbital motion (sorry, Xavier, for making it a separate movement here) of the eletron in the hydrogen atom, the matter was "solved" by QM: the angular momentum of orbital motion in the ground state is zero. Of course, this implies that the notion of Bohr's first orbit cannot simultaneously be correct.

Kai,

You say “the angular momentum of orbital motion in the ground state is zero”. Sorry this is wrong you must read Dirac Work. Dirac P.A.M. Proc. Roy. Soc., A117, 610-624, 1928. Second part A118, 351-361, 1928. In addition there is a correspondence between Dirac and Schrödinger solutions and l equal to zero correspond to l = -1. See also : Total angular momentum and atomic magnetic moments; with G. Lochak, J. Magn. Magn Mater. 65, 99-122 (1987).

It is for me one reason to go back to the Bohr Sommerfeld model and also to find very bad the teaching of the Schrödinger equation mathematically the product l(l+1) give possibility to consider negative values of l and this is not discussed.

On the other hand I appreciate your numerous comments.

Xavier, I hoped to make it sufficiently explicit that I was disregarding spin for that part of my answer. I feared a reaction of this kind. 

All I wanted to say is that QM provides a clue as to why there can be a non-radiative ground state for the hydrogen atom. I don't think the Sommerfeld model can account for this. It remains a postulate (which I find "less fundamental" than an equation of motion).

I have btw started reading into your JMMM article but will not be able to progress quickly. 

 concerning l (l+1) , I do not share your concern. When you know what they relate to (solution of differential equations) then we can find out the values of l to be non-negative.

And allow me one further remark: the degeneracies in energy levels of the hydrogen atom are an "artifact" of the Coulomb potential. In nuclear physics, since the (effective) potential is different, even a different nomenclature for labelling the states has evolved, although a common nomenclature would be possible.

Kai: I agree with you “it is surprising to find the angular momentum with negative values”. But before to reject them you have to consider that they allow the calculation of magnetic moments with a discrepancy with the experiments, often less than 1%. As a result we have to understand the meaning of these negatives values.

Xavier> bad the teaching of the Schrödinger equation mathematically the product L(L+1) give possibility to consider negative

But Xavier, this is a consequence of mathematics: L(L+1), with L=0,1,.., is just the allowed values of the quadratic Casimir for unitary irreducible representations of the rotation group. Nobody teaching the Schrödinger equation for rotation symmetric potentials can get around that, no matter how bad the teaching.

Kai,

First of all, Generalized elasticity in for example Cosserat media (generally called micro-structured) is a vehemently revived branch, and fruitful results are expected at least from my limited point of view. We have seen that one advances in one branch inspired another. And you may not notices there is a huge body of literature on stochastic based QM. Did you see the whole picture? or the dead end of the physics? To me, electrons, quarks, etc, and further development like transportation, quantum computer is like a river running in the desert.

Sorry, typo. transportation.

Sorry again, teletransportation. Due to the auto-correct feature (it deleted teletransportation and replaced with transportation)

the auto-correct feature

together with the people who invented it, should have been teletransported to a galaxy far-far away. The word suggestion feature is sometimes useful, though.

Stochastic quantum mechanics is not a beaten track, it is a railway station. All and every physicist already stopped here, but the trains stay docked.

Xavier: I have not understood: negative values of l would result from what exactly? Before I ponder their meaning, Id like to see what theiy're the solution of.

My prejudice with respect to this is well expressed by Kåre Olaussen's statement (not the one on distant galaxies...)

Xavier, thanks for sending me that article of yours. Started reading into it and immediately have one question: in such a mechanical model as I (believe) see it evolving over the first page, How is the issue resolved that the electron-proton system does not radiate? Bohr (& Sommerfeld) needed a postulate for this.

Have read further, without having triedto follow explicit calculations in detail, though. My "second impression" is that there are quite a few hypotheses in this article, which seem to me as totally ad hoc, not compelling and against experimental facts (as far as I can tell). This is particularly the case for the assumptions of spatial densities (as 1/r) for both proton and electron. I think we know too much about these quys (from scattering experiments) to accept such ideas.

While I cannot judge the work I link to below (and definitely don't have the time to study this in any detail) I find more appealing the attempts of analyzing in more detail the Dirac eqns of motion in a way that was already looked into by Schrödinger: the so-called Zitterbewegung. The links below go to two examples of such work. Both are still being cited, so there seems to be an active community and up-to-date insight may well be beyond the contents of these two papers.

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.23.2454

http://geocalc.clas.asu.edu/pdf-preAdobe8/ZBW_I_QM.pdf

Maybe also interesting (I primarily have Fushan in mind here): ideas about interpretations based on the Klein Gordon equation.

Wil continue to follow this loosely, but have to focus on other things for the days to come...

http://arxiv.org/abs/1211.4645

Article A Novel Interpretation of the Klein-Gordon Equation

Kai: Glade that you appreciate “The symmetry of the motion, the mass and the quantum state” Ann. Fondation Louis de Broglie, 29, 493-512, 2004.

Now it is a long work to fully solve the Dirac equation, Louis de Broglie do it in “L’électron magnétique”, and use 16 pages.

I will not follow David Hestenes who write: My first contention is that the Dirac algebra contains irrelevant features which must be eliminated to reveal its true geometric content. The simplest way to carry out the elimination is to define the \geometrically purified" version of the Dirac algebra, which I call the spacetime algebra (STA), and then compare it with the conventional matrix version. Dirac solutions give very good result in magnetism and the STA seems OK.

Dear all,

I finally think to an experimental result to shed light on the intrinsic kinetics momentum of the electron, the spin of the photon:

Consider the electron coming from infinity and falling under the attraction of the proton. They become a hydrogen atom with the emission of a photon. The photon leaves the atom and takes with it the action h, the energy E = hv and a spin h. By reaction the electron enter in motion of intrinsic rotation or spin with the action h. The energy of the photon leaving the electron induce upon it an orbital motion of recoil and an oscillatory motion of translation. It correspond to the two fluxes of grains allowing to keep the mass of the electron constant see for example “ The wave and the Quantum States or Quantum States and Periodicity” on my site ReaserchGate. As a result the action h of the intrinsic kinetics momentum is share into two equal quantities 1/2h. This approach allows to consider negative values of angular momentum.

 I think you are younger than me and in better position to find further verification.

I will always glad to pursue the discussion.

Cordially,

Xavier

I am no expert on atomic physics, but I think the process of electron capture must have been studied extremely well, both experimentally and theoretically, for many decades. There is a Wikipedia article on this, which ought to be a good starting point -- with guide to further literature. I don't think papers by Dirac and de Broglie necessarily are the best starting points, although they may be of great historical interest.

Dear Kåre,

Thanks for your comment. I will try to go further, but I already published a work on different electronic transition “rare earth electronic configurations and 3d--4f transitions”.

My main interest is to inform of some aspects of the physics that are sometime ignored. It is the case of the Landé splitting factor g = k/(k+1/2) given by Dirac in Proc. Roy. Soc. A118, 351-361, 1928. You will find it page 361.

Now k is the angular momentum taking positive and negative values. The good comparison of the theoretical magnetic moments with experimental values in my opinion establishes the correctness of the g factor and the existence of the negative as positive values of k. See “Total angular momentum and atomic magnetic moments” J Mag. Mag. Mat. 65, 1987, 99-122, with Georges Lochak.

Thanks again

Sincerely,

Xavier

I am answering the main part of your question. I am not very familiar with the given example in support of the question.

According to Racah's theorem, number of quadratic casimir invariant operators is equal to the rank of the group. The spin group SU(2) is a semisimple Lie group of rank one. There is just one casimir invariant operator of SU(2) which is J2=Jx2+JY2+Jz2. It also turns out for the SU(2) case, that Jz commutes with J2.  Therefore eigenvalues of J are conserved quantum numbers and states are simultaneous eigenfunctions of J2 and Jz. This is explicitly written as |1/2 1/2> and |1/2 -1/2> . This is how I visualize two types of spin of an electron namely spin up and spin down.

Dear Biswajoy,

If I understand your explanation the two types of spin belong to the “ns” states. In fact experimentally the “ns” states correspond to just one kind of spin. You can find more result in the paper: “Total angular momentum and atomic magnetic moments” J Mag. Mag. Mat. 65, 1987, 99-122, with Georges Lochak.

A possible theoretical conclusion is that considering the proton the electron turn on itself only in one sense. In addition we have to consider a translation motion and the classical orbital motion see my paper on my ResearchGate site “Quantum State and Periodicity”

Yours

Xavier

Xavier:

For an understanding of the basic nature of the electron, see Section 1 of www.tachyonmodel.com.  Insofar as a very basic description of its stable state in the case of the hydrogen atom, see Section 8 of the same page. 

This does not help with the Dirac model or multi electron atom, as yet, but it does provide a different view of the basic nature of the electron.

Dear All: The notion of spin is closely connected with that of space. The space exist according to the objects that we consider. As soon as there are two objects they must be in motion between them, otherwise there is just one object. As the physical law does not depend of the place of observation, we have to suppose that exchanges of mass in the form of small grains induce the motion. Then consider the motion between the electron and the proton, the angular momentum involves two directions of the space but there is a third direction perpendicular to the two others. These three directions were took into account in Dirac’s theory. If the space is isotropic the action describing the motion must involve all the three directions of it, thus equally the quantification. Then taking into account the relative position of the proton, the electron has a motion of own rotation or spin, and it has also a motion of own translation, otherwise the space will have just two directions, the both with a total action of one unit “h” of Planck constant for the “ns” quantum state. When a quantum state takes place the electron arriving of the infinity loses a photon.  By reaction, due to differential variations of mass this action induce the angular motion but also a motion of translation. These both motions are quantified by an integer number of “h” units. Simultaneously the action of the both own motions, the spin and the translation, is share into two equal parts that is “h/2”. The additional quantum to a “ns” state, has two possibility to be added: to the rotation or the translation. This explains the splitting of the “np”, “nd” and “nf” shell into two subshells and the spin notion has not this role.

Visualize spin as a tiny charge (10^-18 cm diameter)  revolving in a Compton wavelength circumference orbit. Calculate the current and you will find that its magnetic moment is the Bohr magneton, identically.   You can also work out the angular momentum, the mass-energy, and the origin of the de Broglie wavelength from this model, and more.  See www.tachyonmodel.com. (Yes, my web page.)

Feel free to criticize it.

Dear Ernst: Your approach of the spin is not exactly a good answer to the explanation of the division into two subshells of the shells “np”, “nd” and “nf”. Indeed the spin was propose to explain this property and the observation of the half integer numbers in the magnetic moment or in the Zeeman effect. As I propose in fact this property is the result of the rotation and translation motions of the electron in interaction with the proton or the nucleus, see “Quantum State and Periodicity”. Yours Xavier

Dear Kai, On the Feb 8, 2016 you wrote: “All I wanted to say is that QM provides a clue as to why there can be a non-radiative ground state for the hydrogen atom. I don't think the Sommerfeld model can account for this. It remains a postulate (which I find "less fundamental" than an equation of motion).”

If the physical laws do not depend of the place of observation we have to suppose that exchanges of mass in the form of small grains induce the motion, according to this approach the exchange of grains mass are going in such a way to keep stable the atom and there is mainly no radiative effect, see “Quantum State and Periodicity”.

Now there are directions of interaction leading to the formation of compounds.

 Kind regards, Xavier

Dear all: To clarify the notion of spin there is an interesting question; why we do not consider the possibility of one additional quantum along the axis of rotation. In the study of “The quantum state and the notions of spin, wave function and action” I suggest that, for the wave function characteristic of the first subshell np1/2, nd3/2 and so on, there is one unit of quantum associate to the radial function. If it is like this then the quantification concerns the momentum and the perpendicular direction.

As a result the quantification implies the momentum and the perpendicular direction to it. As a result for the ns states this suggest that already the quantification involves the both dimensions. According to measurements of the magnetic moments, this means that for the ns states there is one quantum divided in two to half quanta, one for the momentum the second for the perpendicular direction.

This completes my post of the April 6.

Dear all: It is evident that the motion of rotation of the electron can be considered into two opposite directions in respect of the proton as verified in the experimental results in magnetism. But the possibility of a quantum along the direction of the rotation seems far from of the understanding of many people. If the space is isotropic the quantification describing the motion must involve all its three directions.

Now the difficulties of the QM concern not only the magnetism. In 1963 Cedric L. Chernick published a paper on “Chemical Compounds of Noble Gas” after a first one in 1962 with sixteen co-authors. I discovers these papers 20years later and up two now this discovery is still not sufficiently known. It is difficult to understand the QM without taking into account the importance of the rare gas compounds, it is the capacity of the xenon shell of the rare earths which play a role of bond in the high-Tc superconductors as La2-xBaxCuO4 and several others (See for example About copper valence and superconductivity, X oudet and G T Bhandage and the comment on march 15/17 of “Atoms and Crystal Structures)

 Dear Xavier:

I believe I was a little careless in my previous comments.  So, let me start over.

First of all, it should be noted that the Schrodinger Equation has been around for 91 years now.  The Dirac Equation has been around for 89 years.  If I understand your question correctly, I claim that as extraordinary as these models are, if after all these years a satisfactory explanation for the various orbital differences has not been obtained from them, the likelihood of anything new being obtained from them  now is near zero. Nothing else magic beyond what they have already provided already is going to happen.  Any more work here is a waste of time.

If you want answers, you need to obtain them yourself from fundamentals.  You need something more basic to work with, like, say, the origin of the electron's de Broglie waves.  That is what I was so clumsily trying to get at with my previous quick and dirty web page comment.  It was most unprofessional.

But to continue, you will get nowhere if you don’t understand in detail what an electron is in the first place, and this includes the physical origin of its magnetic moment, angular momentum, and even mass.  But more important, you need to understand what the physical mechanism behind the de Broglie waves is.  When I say physical origin, I mean just that, and not a few vague, pretty equations.  Without these physical origins, you are wasting your time.

Quantum mechanics provides a high level view of the interactions of the electrons within in the atom, but not the physical origins of the internal parameters of the electrons. 

Also, the Standard Model provides none of these physical mechanisms.  It provides a few vague, pretty equations, however.

Section 1 of the web page I mentioned provides the fundamental origin of a vortex electron's magnetic moment, its angular moment, mass, and de Broglie waves.  That is, it will tell you what an electron really is, and how it interacts with other individual electrons.  There are only 11 simple equations in the entire electron section (Section 1), but I believe you will find the 10 minutes you spend reading them to be worth your while. 

The web page, again, is www.tachyonmodel.com

Also, as to how the vortex electron and its Compton wavelets interact with a nucleus, I describe a possible model in Section 6 of the web page.  Granted, that is a little long, but you might find it worth taking a quick look at, especially the physical reality behind some of the fine structure relationships.

Let me emphasize that there is no claim that this will immediately provide the differences in the orbitals at this stage.  However, it provides a starting point for anyone wishing to extend the methodology so as to find the answer you are looking for.  Specifically, it may help provide information as to why the Schrodinger and Dirac Equations behave the way they do.  It will likely be a long road, however.

I would like to note that I, myself , would like to know why the Schrodinger and Dirac Equations behave as they do.

As I previously said, feel free to criticize these models provided you are very specific as to what your issues are.  that way, I can fix them.

Dear all, there are several questions or projects on ResearchGate where the problem is to understand the QM. There is a difficulty to find the solution since for long time many people have forget to learn the old approach of Bohr and Sommerfeld. At this time according to the attraction between the proton and the electron, the electron describe an elliptical curve around the proton considered as fixed. The quantification is attributed only to the magnetic moment. The difficulty encountered with the half integer number of the momentum were attributed to the rotation of the electron around itself, the spin. With his equation Dirac found half-integer value of the momentum. But it is not evident that this result find its origin in the own-rotation of the electron or spin. Consider now the old model and suppose that the fundamental quantum of action h quantified all the possible direction of a motion. Now at the differential scale any motion have two possible directions one giving the rotation the other a motion along the perpendicular to the rotation. If the quantification concern the three equivalent directions of the space, then the quantum of quantification is divided into two equal parts for the two possible directions of the motion at the differential scale of the space. This hypothesis give a different explanation of the half integer number to that of the classical notion of spin.

Dear Ernst,

Article A Physical Model of the Electron According to the Basic Stru...

Dear Albert:

Thanks for the paper. I have tried to download it, but it has this blue tag that floats  around on it making it difficult to read, but I hope to slowly get through it this week, depending on work schedule.  Looks most interesting.

Dear Ernst,

Besides the article written by Stoyan Sarg (see its figure 5 with helicoidal movement) easier to download, I've found another article which might discuss same idea with trapped photon (see 2nd attachement).

What I do not understand if we go after photon trapped, where does the mass of the electron comes from since photon are mass-less ?

http://vixra.org/pdf/1104.0051v1.pdf

http://gsjournal.net/Science-Journals/Research%20Papers-Quantum%20Theory%20/%20Particle%20Physics/Download/5715

Dear Albert:

First of all, thank you for the articles.

Second, to try to answer your question:  If photons' trajectories are bent by gravitational fields, that implies that the photons have mass. (Look up gravitation lens on the internet.)   It is a little difficult to say how much the mass is based on experiment.  However, in the case of a Compton photon. it is reasonable to assume that this has the equivalent mass of an electron.  Mind you, this is an unproven assumption, but it is still a good starting point for investigating the nature of the electron.  Or, to put it another way,  it is a good working hypothesis.

Dear Ernst: You tell “If photons' trajectories are bent by gravitational fields that implies that the photons have mass.” If I am right it seems to me that Einstein got his Nobel Prize with the bent of the light in the neighborhood of the sun. But the photons are still suppose do not have a mass. Am I right?

Dear Ernst:

If we assume the trapped photon to generate the electron, what is the root cause or why such trapped photon would self-generate the known negative charge of the electron ?

Dear all: 

About Einstein’s Nobel Prize:  It was for the photoelectric effect.  However, the really impressive one was general relativity.  That was “proved” by the bending of light from several stars close to the Sun.  However, I question if that was actually gravity, or was the light being bent by the solar atmosphere.

In any case, if I had nothing else to go on, I would maintain the hypothesis that the mass is the same as the energy and, thus, it will be attracted by gravity in the same manner.

But then, there is the Mossbauer effect (umlauted o).  Look it up, especially the Wikipedia article.  A specific experiment performed using it back in the 1950s with a narrow band gamma ray that was directed downward changed its frequency as it descended down into the gravitational field because of its interaction with gravity.  I believe the experiment agreed with theory insofar as the frequency shift because of gravity was concerned.

Hence, it is a safe bet that the mass of a photon is the same as its energy.

Insofar as Albert’s question is concerned, and to take it a step further , I haven’t figured out what the difference in an electron and a positron is, nor why the wavelets interact with the charges, nor have I figured out why the charge maintains its curved orbit.  In fact, attempting to sort these things out might well result into a slow descent into insanity.

All I can say is that I have a model that produces a clue as to how particles work, but this model still has a long way to go.  Remember that the very simple Bohr atom preceded the more complicated Schrodinger Equation and the Dirac Equation.

 Dear Ernst: Thank for you answer with many important points. But we are far from the notion of spin and what give the two sub-shells which are supposed to be the result of the two kinds of spins, a point that I do not shared. 

Dear All: Among the properties of the Quantum State there is one not often underline: between the two sub-shells there is just one quantum of difference, apparently never more. In fact when we study the transitions 3d -- 4f, (see Bonnelle and Karnatak J. Phys. 32 C4, 230-5, (1971) and my work Rare earth electronic configurations and 3d 4f transitions) it appears in the spectra some additional emissions call “S” difficult to explain. One possibility would be the excitation of states with two additional quantum on the first sub-shell 4f5/2. Such states if it is the case are unstable thus the additional structures or emissions. In my opinion this property is deeply connected to the origin of the two subshells and the understanding will clarify the Quantum State.

Dear All:

I finally had some time to look at the papers that have been sent to me.

To be frank, they are like all other electron papers, where the authors think that complexity will solve the problem. They all dream of finding another Dirac or Schrodinger equation by adding complexity. They are all doomed to failure.

There are two approaches to understanding the electron: There is the Alfred Lauck Parson toroidal ring model of the electron from 1913, and there is the very simple revolving charge approach I took.

You need to go to simplicity if you want to understand the electron.

From the comments on this blog, one thing is certain: No one has bothered to look at my web page, where I use utter simplicity to solve the problem of electron spin. It appears that I may have wasted my time here.

However, in the event someone wishes to understand what an electron really is (along with revolving charge models protons, neutrons, deuterons, and mesons), once again my web page is:

www.tachyonmodel.com

In the event anyone bothers to look at it, they will find that the physical origin of the electron's Bohr magneton is derived in three simple algebraic equations. The physical origin of the mass energy is obtained in two trivial equations, and the physical origin of the angular momentum is obtained in 3 easy equations. The physical origin of the de Broglie wavelength is obtained in 3 easy equations.

I.e., you can describe the electron in a total of 12 easy equations.

If you read this, you will understand what electron spin really is. That won't provide a direct physical answer to Xavier's question. It does not directly explain how an ns state differs from an np, nd, or nf state, or but it should provide a good starting point for an investigation that will help visualize the two kinds of electron spin.

If you look only at complex equations, you will get nowhere.

Dear Emst and All: Considering the last comment of Emrst without helping progress on the question “How do you visualize the two kinds of spin of an electron?” I was surprised working on the following: The study of the equation of Dirac gives two kinds of solutions giving the two subshells observed after the ns subshells with the correct number of magnetic states and sheds light on the notion of spin that is the own rotation. The ns states belong to the second kind of solutions. The determination of the values of the magnetic quantum number m shows to have the following limits

For the first sub-shell -l -1 < m < l+2

and for the second -( l-2) < m < l+1

Then the angular momentum of all state is (m + ½).

This shows that there is always the same number of negative and positive values of the angular momentum, but the contribution proportional to ½h has always the same direction.

This half contribution is to associate to the own rotation or spin, but this contribution is always positive. Accordingly there is just one kind of spin but two different sub-shells.

You can work out all these solutions with the book of Louis de Broglie “L’électron magnétique, Théorie de Dirac”

I will appreciate feedback, Xavier

If all you are looking at is Dirac (as extraordinary as it is), you will die someday without finding your answer. (May you live long, however!) Dirac has been around for some 90 years and everyone and his brother has looked at it for more magical results during that time. So, ask yourself if you have more insight than the hundreds of researchers in all that time. The likelihood is neither you or I or anyone else on this blog will get any further than those past researchers.

Dirac is a marvel. It derives the correct expressions for the Bohr magneton and the electron's spin angular momentum from some relativistic matrices. But it does not provide a physical mechanism that generates these parameters.

If you want to make a better gasoline engine but don't know how a gasoline engine works in the first place, how can you succeed? Or, if you prefer to be environmentally friendly, how do you make a better electric motor if you don't know anything about electricity?

You don't know what an electron is, and you have no idea of how it physically generates a magnetic moment, a spin angular momentum, or a mass. You haven't a clue as to what a de Broglie wave really is. How, therefore, can you analyze spin?

Remember the philosophers that debated Galileo? They refused to look through the telescope at the moons of Jupiter because their philosophy said there were no moons there!

This blog has been running for almost 2 years now, you have made no progress, and you are getting nowhere faster and faster because you have refused to look through the telescope.

Learn what an electron is! Look through the telescope at tachyonmodel.com . I challenge you to show me that it is incorrect. Then, use that along with Dirac and you may possibly get somewhere.

Ermst: There is let me say an almost missing link in the teaching in science and probably in many others place “The Critical sense”, your attitude is typical of the absence of critical sense, there is no place for research with people like you. You can be good in mathematics but not in physics, in my opinion it was the case of Dirac.

Dear all: To critic Dirac I suggest you think about the following:

We know that Dirac has transformed the Klein Gordon square operator as a product of two linear operators. It is evident that the square operator work for positive or negative charge associated to positive energy or negative energy. With Dirac it has been supposed that one of the linear operator is for the negative charge and the other for the positive. It is surprising that the same operator is use for two different particles. Well, but at this time the variation of the mass giving two flux of matter between the proton and the electron was not considered. In “The symmetry of the motion, the mass and the quantum state”, I explain that: “The hypothesis of exchanges of matter divided into two fluxes of opposite directions brings a simple answer. Indeed each flux is characterized with a direction of the speed of propagation of the grains, they determine the masses active and inert, thus the energy must be considered as positive or negative according to the flux. The classical equation corresponds to the positive flux and the other equation to the negative flux.”

All this discussion leads to consider the motion of the electron as a reality close to the model of Bohr Sommerfeld.

Dear all: Many thanks for your interest in this question. Recently, (9 days ago) your comments lead me to study the limits of the quantum number m. I underline that the half quantum contribution, ½h to the angular momentum, has always the same sign. It seems to me that this strongly indicate that the notion of rotation does not explain the two subshells. Of course not all of you probably share my approach, it always like this in research and it is natural. The discussion allows to clarify our point of view.

I am now in my 82 year and it becomes often difficult for me to follow you. This to say that I am progressively becoming really retired.

Tanks again Yours Xavier

Dear All: In my previous comment I show that there is just one direction of the half angular momentum ½h. This allows an explanation of the negative values of the quantum number k. It was supposed to be the angular momentum, but as far I have understand the equation of Dirac, it must be considered as the differential equation to find the conditions of the variation of the mechanical action, that is of the motion of the electron in interaction with the proton. This motion of course concerns the three directions of the space. Thus the intriguing negatives values of the quantum number k has to be interpreted as a negative action in comparison of the halves contributions always positive for the angular momentum but also for the momentum along the axe of rotation.

Very interesting!

First Thank to you Nafaa Chbili, James F Peters, Jonah Lissner, Amitabh Saxena, José Juan Peña Leal, and all of you for your interest in the question, my comments and your recommendations.

I wish now to underline one aspect of the action. Thus it is the action which is quantified.

Now according to Einstein “physical laws should be independent of the place of observation”, the exchange of mass in the form of very small grains, seems to be the alone way to have the same law for the interaction if we consider the motion from the proton or from the electron. On the same way of approach it must be the mass which induces the orbital and spin motions, ­“see: The quantum state and Periodicity”.

Thus whatsoever are the interaction in an atom with several electrons and belonging to a crystal or a molecule, the variations of mass allow to keep constant the action along the trajectories of the different electrons. As a result the quantum properties of the different atoms put in view in the periodic table.

Xavier

Dear readers: in a comment for the question “Can the notion of field be immaterial?” I write:

Let me consider the balance between the grains mass of the two fluxes: they concern grains of the both motions: the rotation and the linear one perpendicular to the rotation. To have a motion the sum of the grains entering and leaving one motion must be supposed different of zero, positive for one and equal in absolute value for the other but negative, otherwise the motion will not exist.

This explains the angle between the celestial equator plane and the ecliptic plane. Such property is well established in QM, and verified with the measurement of the magnetic moment of several compounds, see for example “Total angular momentum and atomic magnetic moments” with Georges Lochak.

As already underlined it is to have forgotten the linear motion which forbid to understand the halves quantum numbers ½h. This remark leads to the conclusion that, according to the exchanges of grains mass, a motion between two objects implies a rotation and a linear motion.

Dear All,

With the hypothesis of small grains mass exchanged between the electron and the proton, I propose to work with the Bohr-Sommerfeld approach and to suppose that the quantum h which imposes the quantification, is divided into two parts one for the angular momentum, the other for the linear momentum along the perpendicular direction to the motion of rotation see “The quantum state and the doublets”. Like this the three directions of the space are quantified. This in fact is different from the gravitation which was use in the Bohr-Sommerfeld approach, but the motion of an electron around a proton has no reason to be the result of the gravitation like the motion of the planets around the sun.

I will be glad if this remark can be useful to clarify the quantum state and shed light on the gravitation.

Dear All, since three years I participle to ResearchGate, I have proposed to consider that there is an interaction in volume between the proton and the electron. It is to explain the half quantum numbers well experimentally observed. See “Total angular momentum and atomic magnetic moments" and "The Garnet Ln3Fe5O12”. The quantification via the quantum h thus leads to divide h into two equal parts see “Quantum state, the magnetism and the rotation”.

We can consider the same interaction for a more complex atoms; this also leads to suppose the same interaction with the neutrons and for the gravitation.

In fact we have always considered directional interaction, is it justified? If there are experiments in contradiction with this approach I will be glad to be inform.

Dear All,

Let me introduce here a further comment of the one that I propose in the question “Is it possible to observe the rest mass of the electron?” It seems to me that we have a very important result about the half quantum number.

Dennis Allen in its paper “Foundation of Science, November, 1999,V2 N4in section Spin page 7 wrote: The rest mass of an electron is almost equally divided between its electrostatic and magnetostatic energies.

This of course gives Magnetic moment of one half Bohr magneton for the first quantum state.

As a result if Bohr had suspected the sharing of the electromagnetic energy he will find the half magneton and the need of an additional h/2 quantum unit as introduce by Uhlenbeck and Goudsmit would never been suggested!

This also gives an answer to: “The one question science cannot answer: What is the magnetic field made up of?” of Emmanouil Markoulakis. In a different approach I also propose this kind of sharing with the hypothesis of small grains mass see “The symmetry of the motion, the mass and the quantum state”.

Now, when we read the paper of Dirac giving its equation it is clear that Dirac does not suspect that the electromagnetic energy can be divided into two equal quantities. Indeed is was surprised to get an electric moment.

In fact the study of the equation of Dirac gives two kinds of solutions with the two subshells observed after the ns subshells and the correct number of magnetic states. The ns states belong to the second kind of solutions. The determination of the values of the magnetic quantum number m shows to have the following limits

For the first sub-shell -l -1 < m < l+2

and for the second -( l-2) < m < l+1

Then the angular momentum of all state is (m + ½).

This shows that there is always the same number of negative and positive values of the angular momentum, but the contribution proportional to ½h has always the same direction. This contribution is always positive.

These result confirms that the rest mass of an electron is almost equally divided between its electrostatic and magnetostatic energies.