Dear Behnam: Thank for your answer. Before to discuss Dirac ’equation it is interesting to underline that with the magnetic momentum to quantify the space, it seem that we avoid the problem of the acceleration. Indeed the speed for a circle is a constant and this approach can be appear as sufficient. But why the quantum property of the space would just concern two dimensions of the space. If we suppose as I have suggested in “Quantum state, the magnetism and the rotation” that the axis of rotation perpendicular to the plane of rotation is also involve, then the action is involve and is it possible to understand the halves integer momentum in Bohr magneton. The quantum induce the quantification and is divided into two parts one for the momentum the other for perpendicular direction. This allows to understand expression of the g factor given by Dirac and the negative values of k. Furthermore this allows to use the classical approach of Bohr Sommerfeld.

Dear Behnam: I apologize for my poor comment. The question is in fact 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.

Dear Behanam: Thank for your comment, 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.

Dear Behanam: The mathematics are a language that we use in many place and in physics too.

But the physics is not a result of the mathematics. In the following you can have a feeling of my approach.

In January 2016 I ask the following question “How do you visualize the two kinds of spin of an electron?” In last April I post the following comment:

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.

Dear Xavier, and Behnam,

First of all, Behnam, to indicate books is not a very useful procedure, people who ask questions want an explanation of a couple of lines, not to read a whole book, not even a whole article. They want an answer to the specific issue not clear to them.

Now, I am repeating here the last comment of Xavier, there are a couple of conceptual problems, and I hope to bring some order.

"The notion of spin is closely connected with that of space."

No, Xavier, it's not exact. The spin is a retivistic effect which is not sure that means a rotation of the particle around its own axis. Dirac obtained mathematically the concept of spin, not in connection with, and not looking for, a rotation of a particle around itself.

"The space exist according to the objects that we consider."

Yes, this is a profound statement.

"As soon as there are two objects they must be in motion between them, otherwise there is just one object."

Where from you took this? Two objects may be at rest, one with respect to the other. However, they may have all sort of physical properties which differ.

"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."

Where from you took this? An electron moves around the nucleus, but there is no particle exchanged between them. As long as the electron moves on a certain shell and its quantum numbers don't change, the QM doesn't tell of any particle exchanged between the electron and the nucleus.

"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."

Which two directions Xavier? In classical mechanics, a rotating object has an angular momentum perpendicular on the rotation plane. But the electron on the shell n=1 has NO particular rotation plane, as it has NO particular direction of the orbital angular momentum. Moreover, it has no particular direction of the spin.

You can produce a beam of electrons - outside of the atoms - and pass them through the Stern-Gerlach apparatus. From the two exiting beams you can retain one. It will have all the electron "polarized" with spin oriented in one and same direction.

But, to your attention: NOBODY guarantees that BEFORE the Stern-Gerlach apparatus the electrons were polarized.

"These three directions were taken 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"

NOT TRUE! The spin of the electron doesn't idicate a rotation around itself. Trials to explain the magnetic moment as coming from a charged sphere rotationg around itself, failed.

"and it has also a motion of own translation, otherwise the space will have just two directions, both with a total action of one unit “h” of Planck constant for the “ns” quantum state."

Sorry, this is quite an "incoherent" statement. The electron in an atom doesn't undergo translation and that has no connection with the dimensionality of the space. But the electron has an "orbital" movement - maybe you express yourself incorrectly, naming this movement "translation".

"When a quantum state takes place the electron arriving of the infinity loses a photon."

You want to say that when we make an electron and a positive ion to collide, the absorption of the electron by the ion results in the emission of a photon. Also, the absorbed electron occupies a shell around the nucleus - a certain quantum state.

"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."

There is no translation inside the atom, as I already said. The difference between the total initial energy of the electron and ion (including their kinetic energy), and that of the final atom, flies away in the form of a photon. What yes is, the electron das an "orbital" dynamics around the nucleus.

"Simultaneously the action of the both own motions, the spin and the translation, is shared into two equal parts that is “h/2”. "

I am sorry, there is no such scenario. The scenario is much more complicated. When the electron APPROACHES THE NUCLEUS, its motion can be approximated by a plane wave. The plane wave may, in turn, be represented as a superposition of states, each state with a definite angular momentum. Therefore, in the atom, the electron would be initially in a superposition of eigenstates of the orbital angular momentum. However, a series of photon emissions would follow, and the electron would finally settle on the lowest available energy state.

This is all the story.

The spin angular momentum is equally as fundamental and important. When Planck’s constant appears the topic is quantum mechanics.

Dear Sofia: Thanks for your interesting comment. You say “NOT TRUE! The spin of the electron doesn't indicate a rotation around itself. Trials to explain the magnetic moment as coming from a charged sphere rotation around itself, failed.” I agree but for many people the spin is still an own rotation. So when I say "The notion of spin is closely connected with that of space." I mean the rotation it was first supposed to be a rotation.

So let me say like this “the notion of rotation is closely related to the notion of space!” Do you agree?

Now, how can be two objects without possible interactions than that of their existence? Do we must consider them without relative motion or not? If we consider a motion does this motion is only in a plan of rotation or do we have to consider a motion along the perpendicular direction of the plan of rotation? Kind regards Xavier