Hi

I did not get it. In what sense all quantum states?

Dear Cesar: Bohr Sommerfeld approach gives just half number of quantum states. In particular this model was unable to explain the two subshells np1/2, np3/2, nd3/2, nd5/2 and so on, observed in spectroscopy or in magnetism where the magnetic moment are a half integer number of Bohr magnetons. Dirac according to his equation finds all the states. I underline several aspects of this equation in the question “How do you visualize the two kinds of spin of an electron?”

Now what I wish with this question is that even if some people are critical in respect of the special relativity the Dirac equation is really a good application of it. There is nevertheless a difficulty the variations of speed imply variation of mass according to exchanges of grains mass not to the law of addition of speed. Like this all variations in the space time must be considered at the differential scale as it is in Dirac equation.

Let me underline now that in the introduction of the special theory of the relativity Einstein “wrote physical laws should be independent of the place of observation”, this conclusion of Einstein implies that the mass is built with very small grains mass moving with the speed of the light.

The Dirac Hamiltonian can be shown to be a Hermitian operator on the space of square integrable functions, so it defines a unitary evolution operator.

That's what it means.

Lorentz transformations of the coordinates imply that the function that satisfies Dirac's equation, transforms in a certain way, that's all-it's not a scalar under Lorentz transformations. No problem.

The statements about the mass, are, simply, wrong.

Stefan: 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.

Hi Xavier,

My apologies, but I must admit I may have not understood the main point in this thread maybe. Nevertheless I would like to mention that with hindsight, deducing the Dirac equation "accidentally" gave spin-1/2 as a side effect, whereas Dirac's original intention was to reach to a equation that treats space and time on an equal footing, which is a necessary ingredient for a manifestly covariant equation.

Nevertheless -- historical development aside - - let's not mix up spin-1/2 (and spin in general) with particles/antiparticles, as you mention both topics. In the end, spin is the consequence of the fact that in QM one is interested in unitary representations of (Lie) groups, in this case the rotation group. More strictly speaking, spin naturally emerges when one considers (topologically caused) projective representations of the rotation group, or put alternatively, unitary representations of the universal covering group of the rotation group. Therefore, spin itself is not a relativistic effect, although admittedly some older textbooks still state this myth.

Antiparticles however are a true relativistic phenomenon, as from a mathematical point of view, in relativistic QM, they emerge naturally from either the fact that the time derivative enters quadratically (e.g. Klein-Gordon equation), or, alternatively, when getting rid of the second derivative, one only is able to do so by doubling the number of degrees of freedom (e.g. Klein-Gordon equation in Schrödinger form).

Maybe you could again clarify what your main statement is?

Best regards

Oliver

Dear Olivier,

The main point concerns Dirac equation. This equation can be considered as the way to find a generating function of the action. This role is that of the wave function, but the label itself is misleading.

The difficulty is that in different aspects this equation is not correctly understood. Dirac find the twice the number of stationary states given by that he call “the point charge theory”. As a result he supposed to have find the spin. But if the total angular momentum is a half integer number of Bohr magneton this does not proof that it is the result of spin.

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 result is established with the comparison of the theoretical and experimental magnetic moment See “Total angular momentum and atomic magnetic moments” J Mag. Mag. Mat. 65, 1987, 99-122, with Georges Lochak or on my ResearchGate site. In this work we use 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.

This half contribution seems to be the own rotation or spin, but this contribution is always positive. Accordingly there will have just one kind of spin but two different sub-shells. In fact this result lead to suppose that the quantum h giving the quantum state is shared into two equal parts, between the angular momentum and the linear momentum along the rotation axe.

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

Dear Xavier,

again I need to apologize, but maybe I still don't get your main argument here. In your first statement you refer to the Dirac equation. Then you actually move on to a concrete example where it might be applied, namely the anomalous Zeeman effect (I assume this is what you mean), but in fact you need not necessarily do so, because the anomalous Zeeman effect is nearly completely explainable in terms of non-relativistic QM. All you need to understand is the addition of angular momenta for the spin-orbit coupling, with a spin-1/2 of the electron, and first-order perturbation theory. To get the quantitatively correct results, admittedly, you manually need to take into consideration the Thomas precession. This, of course, automatically is taken care of by the Dirac equation, it being relativistic from the beginning.

This is standard textbook knowledge, and easy to reproduce for any student. So I fail to see the necessity to read any papers that state otherwise. You should somehow be able to point out the main line of argument in a clear and easy-to-grasp way.

Maybe you are saying that explaining the anomalous Zeeman effect could -- at least in principle, from a purely epistemological viewpoint, and from the perspective of a 1925 physicist -- have been done without spin, and by some other means. And maybe it could at that time, I have not thought this through, but I doubt this very much.

But please bear in mind that in any case:

a) the Dirac equation is not needed for explaining either spin or the Zeeman effect,

and

b) the existence of spin is to be inferred by completely other means, namely by the representation theory of the Galilei group and can be shown to be -- alongside with mass -- one of the two fundamental space-time quantities that give a meaning to the notion of "identical particles", which otherwise would not be possible. This is all very fundamental and completely detached from atomic physics.

Best regards

Oliver

Der Olivier,

As a result of the interferences with very low flux of photons, there is no clear evidence of the wave associated to the photon and the electron. I never share this approach, this perhaps induce a difficulty to get my main argument. It is the reason why I underline that Dirac’ equation is a way to find the action which is measured in the same unit than the angular momentum. This allows to consider the corpuscular approach of Bohr Sommerfeld completed by the work of Dirac.

Now why do you say “I move on to a concrete example where it might be applied, namely the anomalous Zeeman effect (I assume this is what you mean)”? I mention the theoretical magnetic moment compared to the experimental results but not the Zeeman-effect. Nevertheless there is an interesting point The Runge law: see Pashen, F., and Runge C., Astrophys. Jour., 15, 235, 333, 1902; Phys. Zeits., 3, 441, 1902. This law finds the same splitting factor as the one given by the Landé factor use for the calculation of the magnetic moments and calculated by Dirac “Dirac in Proc. Roy. Soc. A118, 351-361, 1928. You will find it page 361”. The rung law is mentioned by White in Introduction to atomic spectra McGraw-Hill Book Company, 1934.

If I understand you: the notion of spin you discuss is the own rotation introduced by Uhlenbeck and Goudsmit supposed to be add or subtracted to the orbital moment. Indeed you write “All you need to understand is the addition of angular momenta for the spin-orbit coupling, with a spin-1/2 of the electron, and first-order perturbation theory.” The electron has a rotation with an additional contribution of 1/2 h to the main angular momentum and the question is to understand why? My previous comment was to try to clarify this point. I do not share the classical hypothesis proposed by Uhlenbeck and Goudsmit. To my knowledge the study of Dirac’s equation allows to understand this important question.

Again: you say “b) the existence of spin is to be inferred by completely other means” so you believe to a given kind of rotation before to be able to understand the experimental results, it is so often like this and I am not too much surprised of your approach but we have to understand if it is just a rotation or something in part different. The equation of Dirac allows to understand this.

Best Regards

Xavier

Dear Xavier,

honestly I am afraid you are losing me nearly completely, but where I think I understand at least what you mean, I have to disagree with you.

I have a problem connecting all your statements about action, Dirac equation, antiparticles, (sub)shells, and Runge law -- yet another piece of history you just throw into the discussion that escapes me completely -- into one coherent message, my apologies about that. With each new reply of yours, you throw just another buzzword at me, and then you think it is somehow up to me to merge this into everything else...but it is all just not getting clearer, and it surely is not how scientific discourse works.

I do not get at all what you mean by this: "As a result of the interferences with very low flux of photons, there is no clear evidence of the wave associated to the photon and the electron." And surely I don't understand at all what this has to do with atomic physics, neither with the Dirac equation.

You were speaking of shells and subshells in a spectral notation so I was assuming you were referring to the (anomalous) Zeeman effect. What else?

And it doesn't actually matter whether you agree with the "Goudsmit-Uhlenbeck hypothesis" from 1925 or not (and by the way it is by no means a classical approach). Meanwhile we have another, much more fundamental way to deduce the phenomenon of spin and understand it on a deep theoretical level. I have hinted at to you how this comes about.

You are stating "you believe to a given kind of rotation before to be able to understand the experimental results, it is so often like this and I am not too much surprised of your approach but we have to understand if it is just a rotation or something in part different. The equation of Dirac allows to understand this." and again, I have to completely disagree. We already *have* fully understood spin, and again I am telling you that it can be inferred in a completely fundamental way that has nothing to do with the Dirac equation -- which in turn adds no further information on its nature. This is not my "belief", but the consequence of some deeper theoretical analysis within some well-defined framework. If you disagree with that, it's up to you to explain what is wrong with that view, where the theory fails, and how it may lead to false conclusions.

All the papers you bring up are around 80-100 years old. The books you mention are about the same age. I am not saying that old papers are always wrong, but I think it might make sense to refer to some modern textbook with a contemporary discussion in order to get an up-to-date overview over matters.

I suggest you ask me a concrete question I might try to answer, otherwise I am afraid I do not know how I could contribute to some deeper discussion.

Best regards and sorry again

Oli

Dear Olivier Thank to have spent time with my comment. We are from different cultures, I get two experimental theses and I need to build my understanding on the experimental results. I work a lot to understand quantum mechanics and I got the chance to work with Georges Lochak Dirac' equation. He was very interested with the theoretical calculation of the magnetic moments which are in good agreement with the experimental one.

I wish you a good life,

Xavier

Dear All,

The fact that Dirac’ equation gives all the quantum states using the Lorentz transformation allows a comment on the question of “Simultaneity in Einstein's Relativity”. Indeed the fact that the photons in a beam of light arrive successively appears as a difficulty for the simultaneity, but the hypothesis of interaction between proton and electron with very small grains mass allows to overcome this problem. According to this hypothesis the motion of the electron in a hydrogen atom is a succession of very small and random steps that we will never observe, thus the orbital motion of the electron seems to follows the laws of the relativist mechanics, but at the scale of the grains mass it is a different thing. Now if we come back to the succession of photons they must be describe as a progressive emission of grains mass and it then perhaps possible to preserve the notion of simultaneity.

To be exact, the solutions of the Dirac equation giving all kinds of energy states of the hydrogen atom are only a formal solution. If a researcher does not rely on a copy of the textbook to solve the Dirac equation of hydrogen atoms, he will find that the so-called exact solution of the Dirac equation of hydrogen atom in the textbook is only a form solution, and this solution does not satisfy the Dirac equation. But in this era, any correct theory of physics is not allowed to be published, presumably because the authority of theoretical physics has gained various interests by erroneous theories. See the reference [15] in the article entitled Pan Quantum Theory (5): Chinese Angular Motion Law and Limitations of Quantum Mechanics.

https://www.researchgate.net/publication/324675743_Pan_Quantum_Theory_5_Chinese_Angular_Motion_Law_and_Limitations_of_Quantum_Mechanics

Dear All:

Let me come back to the transformation of the Klein Gordon square operator as a product of two linear operators by Dirac. In “The symmetry of the motion, the mass and the quantum state” I explain the role of the two operators with “The hypothesis of exchanges of matter divided into two fluxes of opposite directions. 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.”

But the hypothesis of grains mass to describe the interaction implies two motions one for the rotation the other for the translation. Indeed the exchanges of grains mass must concerns to motion to avoid to neutralize the motion. Thus one motion can lose grains mass that is energy and the other gains this energy.

Der Olivier, As far I understand the Dirac equation it include operators Px, Py and Pz. The operator Px and Py give the rotation in the plane Ox,Oy now what contribution for Pz. My approach is that the term Pz contribute to a linear motion along Oz. The purpose of the research is to progress not necessary to preserve the text books otherwise you cannot expect to progress.

Xavier

Dear Xavier,

as you have asked me directly, out of politeness, I feel urged to reply to your question. But I am afraid I am going to disappoint you.

The reason being: I am not sure at all where your question is heading to. When you speak of Px etc., do you speak of the momentum operator (momentum along the x-axis)? Or rotation operator (rotation about x-axis)?

Here's my fundamental problem: I don't see where you are heading to. This whole thread does not make sense to me. Similarly, you have started yet another thread: https://www.researchgate.net/post/What_about_if_the_difficulties_to_understand_the_quantum_mechanics_were_to_consider_that_the_interactions_are_not_directional_but_spherical, where miraculously, again, some similar wordings and phrases reappear.

Compared to other people in this forum at large, I give you a significant amount of politeness, and patience. I surely do not count you as one of those many of the self-acclaimed geniusses that feel an intellectual superiority in them and despise "traditional" scientists. Compared to them, you seem to me a kind, interested person. And if you asked me: "What does the Dirac equation mean? Where does it come from? Are there modern views on it?", I could give you a serious answer. But you don't do that, unfortunately.

From a scientific point of view, I think I must ask you to start studying physics from scratch. I understand that you are very interested in it, but it is the foundational knowledge you are totally lacking. There are so many good undergraduate books to start with.

I cannot see how to help you otherwise.

I am sorry.

Oliver

Dear Olivier,

I do not see where I am “out of politeness”. It is clear, we do not agree each other, so I tell you “The purpose of the research is to progress not necessary to preserve the text books otherwise you cannot expect to progress.” As you read French in my language je dirais pour progresser il faut dépasser l’enseignement de nos maitres. Je ne pensais pas vous parler impoliment.

That I can add, I work since a long time with my colleague of the Fondation Louis de Broglie and they several time told me that de Broglie was questioning the role of the z axis in the Dirac’s equation.

Avec mes meilleurs sentiments

Xavier

Xavier,

here you have now misunderstood me language-wise: "out of politeness" referred to the fact that I felt it is polite to reply to your question, even though I don't see much sense in this thread. It does not mean you are being impolite...hope that's clarified.

I do read and speak French too, by the way. But as I stated and try to convey, it seems the issues we are experiencing with this discussion are not language-related...

Questioning our forefathers and their statements is a totally valid concept. But that is not the issue here. The issue is I simply do not get your point, and in my opinion you are using terminology you do not really understand, and the kind of reasoning you do is not the way you reason in natural science. It does not matter at all what de Broglie said or thought during his lifetime from a scientific point of view.

In order to question something, one first needs to have understood it. Whether one likes it or not, scientific progress is hard work, even if you are a genius. Therefore, again, all advice I can give you is: learn physics from scratch, if you are interested in it, at least to the degree necessary for formulating the respective questions -- otherwise there won't be any serious discussion at all. There is no shortcut way around the hard labour of learning that comes along with it.

Best regards

Oliver

Dear Olivier, with 82 years let me focus your attention on the difficulties of the physics:

In the paper “The Wave and the Quantum State” I wrote in section six: As we have seen repeatedly the notion of point is a significant barrier to progress towards a better understanding of atomic physics. Behind this difficulty there are mathematics prevailing in physics as master while it is physics or in a broader way the experiments that allowed to build mathematics. The result is the difficulties of the Copenhagen school and its members is that they do not takes into account the fact that that any object has a volume and that at the atomic scale the concept of point almost lost much of its value. 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. I hope that these few lines can help researchers. There is a whole field of research that I recommend to my young colleagues.

Best regards