Talk:Exchange interaction

What is being exchanged?

Aren't the "exchange interaction" and "exchange force" in the first paragraph two different things? The first results from the antisymmetry of identical fermions without reference to interactions with other particles. The second results from the exchange of bosons, whether the fermions are identical or not. Maybe the "exchange force" should be discussed elsewhere. —JerryFriedman 19:18, 15 February 2007 (UTC)

I am not an expert on this subject, but I think that the example of a gluon (boson) passing between two quarks is inappropriate for this article. The "exchange" being discussed here is two fermions of the same type trading places with each other. When two atoms are close enough that an electron orbital of one overlaps an electron orbital of the other, the two electrons (of the same spin) can trade places in the overlap and this creates a repulsive force also known as the Pauli force. JRSpriggs 05:56, 16 February 2007 (UTC)

Presently, exchange force is a redirect into exchange interaction, hence there my be a certain amount of overlap of differing material in the article. I will move some of the material into exchange force as its own article, although technically they may be the same thing in some discussions. --Sadi Carnot 16:19, 24 April 2007 (UTC)

Exchange interactions are not just for electrons/fermions

The article gives one the impression that the exchange interaction only is a factor for electrons (or perhaps for fermions) -- it happens for both bosons and fermions, and I'd remove all reference to electrons since it happens for all fermions, not just electrons.

There is an excellent description of this in Griffifths, "Introduction to Quantum Mechanics", second edition, pages 207-210. Basically, identical fermions with overlapping wave functions are observed "further apart" than distinguishable particles, and identical bosons with overlapping wave functions are observed "closer together", where we think of distance here as the expectation value of the square of the difference of the positions of the particles.

Anyway, I think the subject is confusing enough already without making it look like a fermion only phenomenon etc. --Pmetzger (talk) 19:42, 28 January 2008 (UTC)

I'll agree with P. Metzger too, an would like the plurality to return into the title as stated above, e.g.: Exchange "interactions" as well. It will possibly involve some redirectional consequences, but wouldn't so be better spoken of many interactions instead of much interaction? D.A. Borgdorff: 86.83.155.44 (talk) 00:49, 30 January 2008 (UTC)

I'm not sure why it should be plural -- it is a unitary phenomenon, not several phenomena. --Pmetzger (talk) 02:32, 30 January 2008 (UTC)

Incorrect wording on wave function symmetries?

The article says: "For example, the exchange interaction results in identical particles with spatially symmetric wave functions (bosons) appearing "closer together" than would be expected of distinguishable particles, and in identical particles with spatially antisymmetric wave functions (fermions) appearing "farther apart".

Aren't fermion wave functions supposed to be totally antisymmetric (space/orbital symmetry X spin symmetry), and vice versa for bosons? Perhaps the article is correct if you consider the spatial wave function to be a complete wf in Hilbert space, but in that case the wording should perhaps be chosen more carefully. —Preceding unsigned comment added by 93.141.69.128 (talk) 00:11, 23 June 2010 (UTC)

Question on equations 4,5, and 6?

The article's equation (3) adds quantities C, B, and Jex to an energy quantity E0. The integrals C, B, and Jex contain only component wavefunctions phi (nondimensional) and distances, giving them units of some power of distance. Energy seems like it should have a different dimension, unless it is being nondimensionalized somehow.

And on the unperturbed hamiltonian definition: It contains two Coulomb attraction terms 1/r1 and 1/r2 - these should be from each electron to each nuclei, not from the electrons to the origin, right? (Could be more clear) User:Anonymous — Preceding unsigned comment added by 50.160.151.149 (talk) 02:54, 12 September 2014 (UTC)

Missing Ra2 and Rb1 in the equations

In the text is mentioned "electron–proton attraction (r_a1/a2/b1/b2)" but there is no such $r_{a2}$ or $r_{b1}$ in the equations. See in particular the expression of ${\mathcal {H}}^{(1)}$ between equations (2) and (3). Since it is expected that the proton a electrostatically interacts with the electron 2, as well as the proton b with the electron 1, I also expect $-{\frac {e^{2}}{r_{a2}}}$ and $-{\frac {e^{2}}{r_{b1}}}$ terms.

Besides this, in the expression of ${\mathcal {H}}^{(0)}$, we have two more terms $-{\frac {e^{2}}{r_{1}}}$ and $-{\frac {e^{2}}{r_{2}}}$ that I don't understand. Are they the 2 missing terms I mentioned?

Anyway, something must be wrong, at least in the wording, if not in the content. The reference to r_a1/a2/b1/b2 is a broken link, as far as $r_{a2}$ or $r_{b1}$ are concerned.

Check http://phycomp.technion.ac.il/~riki/H2_molecule.html for an equation that includes all these terms without issue with the wording. JPB (talk) 12:11, 6 August 2015 (UTC)

Proposed article re-organization

I think that this article is of very low quality. First, the Hamiltonian given for what is inconsistently referred to as hydrogen or helium has terms with no explanation, taken from a technical document. Then perturbation theory happens. Then the introduction of spins refers to a matrix in perturbation theory!

Griffiths quantum 5.1.2 finds expressions of the mean square distance between two particles with given wave functions for indistinguishable and distinguishable particles. For fermions and bosons this is $\pm |<x>_{ab}|^{2}$ (just $\pm$ some positive term). Thus whenever there are Coulomb-like repulsive interactions, spatially symmetric particles will have a larger interaction energy for given wave functions. Then consulting this source, http://uw.physics.wisc.edu/~himpsel/449exch.pdf, we can establish that electrons of opposite spins are symmetric with respect to exchange of spatial parts of the wave function. There are no big Hamiltonians or perturbative things involved in this explanation. FrozenWinters (talk) 05:31, 26 December 2016 (UTC)

Improvements suggested / needed

I'm not an expert on exchange interaction and I find found article hard to understand. After a bunch of reading different resources, I now get the idea (I think), and I have to say this article was sadly not helpful.

• It should be mentioned somehow that the Born–Oppenheimer approximation is used, and also that spin-orbit etc are ignored. The calculated result (the exchange integral) is from perturbation theory.
• "We assume that Φa and Φb are orthogonal" .... but they aren't, which is why S is nonzero. Weird.
• It's not true that the exchange interaction is the same as Pauli repulsion, which is a somewhat vague term by the way. If I understand right, Pauli repulsion can refer to two different things.

• First is the phenomenon where two atoms resist being pushed arbitrarily close together. If I understand right, this is dominated by the semiclassical Hartree stuff and simple Pauli exclusion principle.
• Second, Pauli repulsion sometimes refers to the electron correlations, i.e. the tendency for like-spin electrons to not be near each other, simply due to the wavefunction antisymmetry. But neither does the phrase 'exchange interaction' refer to these electron correlations!

• Rather, 'exchange interaction' refers to the integrated-out *effect* of those correlations in combination with other forces like Coulomb (in various ways). Moreover and specifically, I find "exchange interaction" seems to often refer to the simplified effective ${\vec {S}}\cdot {\vec {S}}$ interaction that gets set up between the spins of electrons in different orbitals/lattice sites, e.g. as in the Heisenberg model.

Personally I find the Koch book chapter to be most helpful and I think this article could borrow a lot from it. --Nanite (talk) 21:26, 7 February 2022 (UTC)

I added a section to try introduce the topic.

The Mullin and Blaylock ref consider Pauli repulsion to be fermion exchange force.

AFAIK the resistance of say two He atoms is entirely Coulombic: the positive charges of the two nuclei are shielded by the two electrons from each atom so the electrons on each atom only see repulsion at short range. You could also create molecular orbitals for He₂ and fill then with four electrons. Then exchange symmetry would require two in the bonding and two in the anti-bonding (anti symmetric) levels but then you have to make further analysis to understand their collision. So I don't think Pauli repulsion is helpful for that case. Johnjbarton (talk) 16:24, 17 February 2024 (UTC)

Pauli repulsion

I've removed the mention of Pauli repulsion. This is mainly because I could not understand what exactly was meant by Pauli repulsion here, and there were no sources. The exchange interaction arises from the combination of wavefunction symmetry and Coulomb interaction. Is the same true for Pauli repulsion, or is it more purely about symmetry (in which case it probably does not belong here)? I don't think it is completely unconnected (and neither is volume/stability of matter, another term which I removed for now), but some sources are needed to establish the connection. Jähmefyysikko (talk) 14:15, 17 February 2024 (UTC)

Pauli repulsion is just another even more colorful name for the fermion exchange "force".

I added this back in two places with the the Mullin and Blaylock ref; I put the ref in the intro (also) but feel free to remove that one. We could choose not to have this in the intro but Pauli repulsion is (or maybe was ;-) ) widely used in intro Chem classes. Johnjbarton (talk) 16:08, 17 February 2024 (UTC)

FYI Pauli repulsion redirects here. Johnjbarton (talk) 16:09, 17 February 2024 (UTC)

It seems there are two separate but closely related terms here:

• Exchange force/Pauli repulsion. Effects between identical particles due to exchange symmetry/Pauli repulsion alone. Griffiths and Mullin/Blaylock mostly discuss these.
• Exchange interactions. Spin-spin interactions of the form $H=\sum _{ij}J_{ij}S_{i}\cdot S_{j}$, which arise from exchange symmetry in various ways. These are effective interactions, obtained by 'integrating out' degrees of freedom other than spin direction.

Both fit into the same article, but we should be careful to make it clear which one we are discussing at each point. Jähmefyysikko (talk) 17:03, 17 February 2024 (UTC)

Ah ok now that makes a lot more sense. The whole first part of the article was so opaque to me that I never got to the later part. So the effect of exchange symmetry on spin-spin interactions in eg magnetism.

These are both consequences of exchange symmetry (AFAIK) just applied to different systems. In atoms exchange symmetry leads to electron shell and Hund's rules because the overlap term is right across the atom. In metals, the symmetry leads to variants of magnetism depending on overlap.

To me the whole middle part of this article obscures rather than illuminates. What do you think? Johnjbarton (talk) 18:01, 17 February 2024 (UTC)

The E. Koch book chapter listed by @Nanite seems like a good ref. Johnjbarton (talk) 18:14, 17 February 2024 (UTC)

The middle part is certainly not very clear. It is probably some traditional derivation, and not necessarily the best way to approach this subject. Koch is fine as a ref as it is freely accessible, but so far the best reference I found is the Handbook of Magnetism and Magnetic Materials entry Magnetic Exchange Interactions by Ralph Skomski. Jähmefyysikko (talk) 10:09, 18 February 2024 (UTC)