Dear Raman,

Be sure that I will not use too much mathematics. I have tackle the question in the following paper “Quantum States and Periodicity” Ann. Fond. Louis de Broglie 36,137-157, 2011. In this paper I discus the Stern and Gerlach experiments p150-151. You can have it on my ReaserchGate site.

If you do not know yet let me tell you: When Dirac published his equation he has supposed to have find the spin because probably he found half integer values for the angular momentum. But according to the solutions of this equation, it is clear that the “ns” states correspond to just one spin state, contrary to that is generally supposed.

The two sub-shells of the “np” “nd” and “nf” shell correspond to an additional quantum state to that of the “ns” states, with a different number of states. This is exhibited for example with the Zeeman Effect. This is different from the classical notion of spin according to Uhlenbeck G.E. and Goudsmit S., where the spin hypothesis was proposed to explain the two subshells “np” “nd” and “nf”.

This is also established with the calculation of the magnetic moment of different compounds.

Now you discuss the double stern and Gerlach, have you a reference I will be glad to know it.

I wish you good work,

Xavier

sir i didn't  get any help from your paper.

It would be very helpful if  you straightaway answer it.

Dear Raman,

You are working on a subject without a full consensus, so my first reaction was to advise you of the fact that the notion of spin can be discussed.

Indeed 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 of k = l and k = - (l + 1) with k¹0 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 mean 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 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 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.

In my paper with Georges Lochak “Total angular momentum and atomic magnetic moments” J. Mag. Mag. Mat. 65, (1987), 99-122, we give in section 4a discussion on the notion of spin, but in section 2 you will find a lot of mathematics the work of Lochak.

I never work with uncertainty principle, according to my work the electrons follow a trajectory.

You will probably need time to assimilate QM so do not hesitate to question me. It is a difficult subject?

Xavier

Raman Choudhary,

let me try to give you an account within (what I believe to be) the "conventional" understanding within quantum mechanics. "Conventional" is meant to say that it is permissive to assume for the silver atom to possess zero orbital angular momentum and spin 1/2. (In doing so we neglect at least two things: (i) we don't take into account the full character of the Dirac equation solutions but reduce it to the Pauli approximation; (ii) we neglect that the stable isotopes of silver possess a nuclear spin of 1/2, too, i.e. we neglect the so-called hyperfine coupling).

Notice that Xavier has his own notion of how angular momentum should be dealt with in the hydrogen atom. I personally am unable to follow his arguments, though.

In my description below I shall discuss setups with one,  two and three Stern-Gerlach devices, respectively. I do of course assume that these are perfect machines :-) In this sense, the story I will be telling is a so-called "Gedankenexperiment" [yeah, at times we Germans invent teriffic words, don't we...]. To begin with, I will start out pretty much the way you did when formulating your question.

So, consider a setup with just one Sz device. It does split up the beam in two of equal intensity. This is the classical SG experiment. Now add a second SG device for Sz (again). We set it up such as to accept just one of the two split beams, however. What we will find is, that when we let the primary Sz+ beam pass the second SG filter, all the atoms will follow the Sz+ path of this second device. Likewise, when we let enter the primary Sz- guys, they will follow the exit lane of the Sz- channel in the second device, too, without exception.

What would we learn from this? By selecting the first path (i.e. the value +1/2 or -1/2 for the z-component of the atom spin), the outcome of the second measurement of Sz is determined to 100%, and not undecided as with the primary beam where we knew nothing about the spins.

We will (of course) obtain the analogous result if we do the same experiments when turning both SG devices such as to split the beam along the x direction instead of z.

OK, now we turn the second device only into the x direction. You have correctly described what will happen. Upon selecting either the Sz+ or Sz- fraction, the beam will split up in two again, along x. The same would be true were we able to admit both, Sz+ and Sz- type atoms simultaneously. We just get twice as many atoms in each channel.

Admittedly, up to this stage nothing too strange has happened as long as you are willing to see a beam splitting in two as not being strange enough. What is "disturbing" and "quantum" and related to the uncertainty principle will follow in my next post.