Dear SoFia,

The two fermions (neutral atoms, rather than nuclei only) have the same energy. 

Might they interact in such a way that the energy levels split apart as a result of the spin of one flipping over?

Paul

Remi,

the issue is not simple. "Force" is just a word.

The two fermions WON'T come to the same detector. Let's think that we repeat the experiment with many pairs of fermions, one fermion from the source A and one from the source B, and place detectors at different depths below the sources. We would see that the wave-packets of the two fermions get closer to one another as the depth increases. However, at some depth we expect to see that the fermions don't continue to get closer and closer to one another. Of course, because of the destructive HB&T effect.

Though, it seems as if the group-velocity changes. The phenomenon is "as if" there is a force.

UNFORTUNATELY, it's difficult to do such experiments with fermions - maybe you'd take a look at what I wrote at the page of my new project

https://www.researchgate.net/project/It-is-possible-practically-to-perform-the-Hanbury-Brown-and-Twiss-experiment-with-fermions

If the experiment with fermions - e.g. 3He could be performed, it would be very interesting.

With kind regards!

Dear Paul,

The picture with the two fermions, in some more detail, is as follows: the wave-packet of each fermion is not point-like, it may be for instance a Gaussian in each dimension. As a Gaussian occupies a volume in space, it also occupies a volume in the linear momentum space. By the way, because of this spread in linear momentum the Gaussian also increases in volume as the particle moves toward the detector.

This is one of the difficulties in performing such experiments, for avoiding the spread in volume one should work with heavy fermions and this is not simple technically.

Now, since there is a spread in linear momentum, there is a spread in energy. The word "split" that you use is fit for a discrete range of energies, not continuous, as in the wave-packet.

Despite these problems, I don't see any reason why the group velocity of the two wave-packets, change when the wave-packets come close to one another.

Δx Δpx < ħ is not quite correct. Δx Δpx is of the order of  ħ. From the fundamental set of commutation relation of operators (three for position, three for momentum, three for spin) we see that one can specify only one component of spin for each particle. I see that the Z axis has been fixed already. There will be a contribution from spin-spin interaction to the total interaction energy in this case, which is fundamentally related to Pauli's exclusion principle of-course. Various phenomenological expressions for spin-spin interaction are considered in the literature. For a simple interaction term which is bi linear in spins and isotropic under rotation please see, http://www.lassp.cornell.edu/clh/p654/MM-Lec0.pdf (Eqn. 4.1.18) Finally one has to solve the Schrodinger equation of this two particle system, I guess.

Dear Biswajoy Brahmachari,

thank you for the reference. But it deals with multi-particle systems, i.e. more than  two, and in those systems the spins have the possibility to allign their directions to be parallel, or opposite, etc. It's not my case. The spins are prepared in the z direction and the two particle don't interact. Anyway, I sought in that article the order of magnitude of the range of the spin-spin interaction, but I didn't find.

Now, the relation

(1) Δx Δpx < ħ

is what is FORBIDDEN to happen. As far as I know, this prohibition doesn't come from a spin-spin Hamiltonian, but from the antisymmetry of the wave-function at a rotation by 360º in space. I guess that for questioning the occurence of spin-spin interaction the particles have to get very close to one another, and the prohibition of the relation (1) won't permit that.

Of course, it's just a guess, I need orders of magnitude of the range of the spin-spin interactions mentioned in the article. Please be kind and correct me if I am wrong.

Again, many thanks for the material, it widens my knowledge.

(By the way, are you familiar with the Hanbury-Brown and twiss experiment? I would be very glad. The above prohibition appears there as a major issue.)

Sofia 

Thank you for pointing my attention towards the Hanbury-Brown and Twiss experiment. I was not quite aware of this experiment, regards, Biswajoy

Dear Biswajoy Brahmachari,

I highly recommend the reports of the experiments of the group of Alain Aspect, e.g.

QUANTUM ATOM OPTICS WITH BOSONS AND FERMIONS

You can find the article in quant-ph, and if you are interested I can indicate additional references of this group about the same experiments. These experiments are a pearl of beauty and experimental skill, as simple as that.

Best regards,

Sofia

Hi Sofia,

In the present case, HB&T-Effect for fermions will be similar to the Aharonov-Bohm Effect. As the finite packets of He-3 move, their A-vector potentials arising (remember that the perturbation of spherical symmetry of the wave packet arising from He-3's motion will  give rise to A-vector) from the central protons (++) minute deformations will dominate in the direction perpendicular to their directions of motion. Electrons, usually, will remain immersed in the protons field and would not participate in the process. For this reason, when the fermions come closer they will attract each other. As they come closer they will experience vacuum attraction force similar to Casimir force. Casimir force will generate further attraction between the wave packets.Temperature will play an important role in the whole process. At lower temperatures (ultra cold) AB-Effect will be dominant. No such effects will be visible at higher temperatures. What you think?

Best Regards,

Aziz

Hi, Aziz!

Thanks for your explanation, it is interesting. I have to think on it. With the Casimir force I am not so trained, but it seems to me that you suggest a dipole deformation. Do I understand correctly?

If, "yes", why do we need a Casimir force?

With best regards,

Sofia

Hi Sofia,

I view Casimir potential as the positive charge potential arising from the central positive charge of an atom. At very low temperatures, when the thermal photons bombardment on He-3 atoms reduces, potential field from  the positive nuclei extends further. Basically, the field generated by a positive nucleus dominates over the  negative charge field generated by electrons in the atom. Virtual photons generated in the process creates attraction between two neutral atoms. Spatial orientation of the virtual photons is responsible for this attraction. We sometimes call this attraction the Van der Wall's force. At very low temperatures, positive potentials generated by the He-3 nuclei will lead to their attraction. In the process, He-3 atoms will condensate to superfluid state. This effect also goes by the name "order parameter" in condensed matter physics. It is not a dipole, rather, a disk of A-vector perpendicular to each atoms motion. Here, we have both the phi-potential and the A-vector in action. A-vectors make two He-3 atoms participate in the Aharonov-Bohm effect leading to their attraction. In addition, the phi-potential, which generates virtual photons generate the Casimir effect. Casimir effect also leads to  He-3 atoms attraction.  Notably, the abstract charge field arising from the nucleus and the electrons is absent outside the neutral atoms. For this reason, there is no Coulomb force in the whole process.

Hope you find this helpful.

Best Regards,

Aziz

Dear Aziz,

What you tell me is very interesting. But, I think (without being sure) that the two fermions don't even have the possibility to come as close to one another as to generate such processes. Also, I didn't say that we lower the temperature so much as you assume.

Are you familiar with the Hanbury-Brown and Twiss effect? The effect says that the two fermions are simply forbidden by the very wave-function, to get so close to one another.

It seems to me that the picture should be as the one I attached, i.e. repeating the experiment with the detectors placed at different heights, one should find a forbidden region. TheHB&T effect predicts that the forbidden region repeats itself, but my picture shows only the central minimum.

The question is what happens in the forbidden zone? Do the fermions bump away from that zone?

With best regards,

Sofia

The ``forbidden zone'' is, just, that-one doesn't find the fermions there. That's the result of the interaction described by the statement that the particles are fermions. So, either one says that one's describing two fermions, or one can say that one's describing a certain interaction between two particles. It's the same thing. 

Hi Sofia,

Since you acknowledged near the start that the two identical fermions can't reach the detector together, can one approach your question in terms of an elementary scattering approach?

I'm minded of a section in Feynman's Lectures (link below), and have suggested three other links (in no particular order) that might offer an easy route into this approach.

Best - Paul

PS I've got a couple of 'heavier' refs if this approach is of interest. 

http://www.feynmanlectures.caltech.edu/III_04.html

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm

https://farside.ph.utexas.edu/teaching/qm/Quantum/node114.html

http://www2.ph.ed.ac.uk/~gja/qp/qp13.pdf

Hi Sofia,

I've just located the following monograph, which includes discussion of Helium 3 and also scattering.

I haven't checked the (very long) Table of Contents, for discussion of He3-He3 scattering in particular - but it does include references to Hanbury-Brown & Twiss:

Ultra-Cold Fermi Gases – 27 May 2014

by International School of Physics Enrico Fermi Paperback: 933 pages

(27 May 2014) ISBN-10: 1607503182

ISBN-13: 978-1607503187.

It might not be easy to get hold of, although Abe Books lists an expensive copy 

If it sounds promising, maybe someone on R-G has a copy...

Incidentally, in my previous post, immediately above, I was thinking more about uncombined fermions (electrons etc.) rather than neutral atomic fermions. So I hope the ref in this reply should be more relevant to your question.

Paul

PS From the abstract below, perhaps I shouldn't have deleted this afternoon's posting about quantum degeneracy after all. I'll come back to that later.

https://books.google.co.uk/books?id=ZNbvAgAAQBAJ&pg=PA390&lpg=PA390&dq=%22atom-atom+scattering%22+gas&source=bl&ots=e93Hm83lnM&sig=g3WXLlzP5sGpbFGqmbgDZpQm3AM&hl=en&sa=X&ved=0ahUKEwjn79Gy4t3QAhXkD8AKHZtPCosQ6AEINDAD#v=onepage&q=%22atom-atom%20scattering%22%20gas&f=false

https://www.abebooks.co.uk/servlet/SearchResults?n=100121503&sts=t&tn=Ultra-Cold+Fermi+Gases