# Is the Mendeleev table explained in quantum mechanics?

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Does anybody know if there exists a mathematical explanation of the Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the authors present quantum mechanics as an axiomatic system, so one could expect that there is a deduction from the axioms to the main results of the discipline. I wonder if there is a mathematical proof of the Mendeleev table?

P.S. I hope the following will not be offensive for physicists: by a mathematical proof I mean a chain of logical implications from axioms of the theory to its theorem. This is normal in mathematics. As an example, in Griffiths' book I do not see axioms at all, therefore I can't treat the reasonings at pages 186-193 as a proof of the Mendeleev table. By the way, that is why I did not want to ask this question at a physical forum: I do not think that people there will even understand my question. However, after Bill Cook's suggestion I made an experiment - and you can look at the results here: http://physics.stackexchange.com/questions/16647/is-the-mendeleev-table-explained-in-quantum-mechanics

So I ask my colleagues-mathematicians to be tolerant.

P.P.S. After closing this topic and reopening it again I received a lot of suggestions to reformulate my question, since in its original form it might seem too vague for mathematicians. So I suppose it will be useful to add here, that by the Mendeleev table I mean (not just a picture, as one can think, but) a system of propositions about the structure of atoms. For example, as I wrote here in comments, the Mendeleev table states that the first electronic orbit (shell) can have only 2 electrons, the second - 8, the third - again 8, the fourth - 18, and so on. Another regularity is the structure of subshells, etc. So my question is whether it is proved by now that these regularities (perhaps not all but some of them) are corollaries of a system of axioms like those from the Berezin-Shubin book. Of course, this assumes that the notions like atoms, shells, etc. must be properly defined, otherwise the corresponding statements could not be formulated. I consider this as a part of my question -- if experts will explain that the reasonable definitions are not found by now, this automatically will mean that the answer is 'no'.

The following reformulation of my question was suggested by Scott Carnahan at http://tea.mathoverflow.net/discussion/1202/should-a-mathematician-be-a-robot/#Item_0 : "Do we have the mathematical means to give a sufficiently precise description of the chemical properties of elements from quantum-mechanical first principles, such that the Mendeleev table becomes a natural organizational scheme?"

I hope, this makes the question more clear.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
retagged Mar 29, 2016
You might want to try posting this question on theoreticalphysics.stackexchange.com

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Bill Cook
I would suggest rewriting your question in purely mathematical terms. As it stands, it is best asked in the theoretical physics stackexchange (as Bill Cook mentions). But there is indeed a mathematical question here -- but you did not ask it.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Jacques Carette
Jacques, what do you have in mind?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Bill Cook - that's a terrible advice. theoreticalphysics is specifically for research-level questions, while this one seems to be more suitable for physics.stackexchange.com (difference between those two sites is the same as between MO and Math.SE)

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Marcin Kotowski
"closed as off topic by Greg Kuperberg, Mark Sapir, Todd Trimble, Ryan Budney, Alain Valette 30 mins ago" @ Greg Kuperberg, Mark Sapir, Todd Trimble, Ryan Budney, Alain Valette - gentelemen, may I ask you, why did you decide that this is off topic?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
It's off topic because the periodic table is chemistry before it is anything else, and the Schr\"odinger equation is physics before it is anything else. It's a great chemistry-and-physics question, but as a strict math question it's hopelessly arcane. There could be some related math question that's reasonable, but a revision of your question is anyone's guess.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Greg Kuperberg
@Greg: 1) if you ask this "great chemistry-and-physics question" to chemists or physicists, they will treat you as an idiot (and this is what happened to me at theoreticalphysics.stackexchange.com/questions/473/… ). Because they do not understand what logic is. If this were not so, there would not be contradictions between what people write here and what they write there: "Yes, quantum mechanics... – fully, quantitatively, and comprehensively explains all of chemistry..." So I still think that I should address this to mathematicians.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Greg: 2) will the following revision of my question be suitable: "should we treat the postulates of quantum mechanics as axioms of an axiomatic system?"

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Mark: 1) if physicists did not convince me, like Luboš Motl at their physical forum, that everything is "fully, quantitatively, and comprehensively" explained, I would not ask this question here. But their arguments are always so definite, uncompromising, unequivocal (and I would say in some sense offensive, don't you find them so? :), that you begin to think that maybe you read wrong books, and if you ask mathematicians who are interested in tags like "quantum-mechanics", they will give an explanation, which could be verified (as this usually happens with mathematicians).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Mark: 2) concerning this: "There is no option "question does not make any sense" in the list of closing options." -- It is not me who pretend that quantum mechanics is based on "postulates", so this is pulery mathematical (if you want, logical) question, if this pretension can be considered as a presentation of an axiomatic theory. Doesn't this make sense?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Mark: from this discussion it follows that there is a qualitative difference between mechanics (as a part of differential geometry) and quantum mechanics (as a part of what you say): in the first case this is an axiomatic system, while in the second one this is not. And this is not obvious for non-specialists like me. So, when appealing to colleagues, I thought I could count on their understanding, since they at least are able to understand my question. But what I see here from moderators is not called understanding, this much more resembles rudeness ("hamstvo" in Russian, is it? :).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Mark: I guess, this means that you do not think I am a mathematician. You can check this by the link I gave at my page here: mathnet.ru/php/… Is "Doctor of Physical and Mathematical Sciences" enough high level for you?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: There is certainly a germ of a legitimate question in what you asked, but as Jacques Carette already pointed out, it should be written in purely mathematical terms, i.e., it should probably ask whether certain facts about eigenvalues and eigenfunctions of a certain Schrödinger operator have been proved (see, for example, the slides Terry Tao refers to). After you ask a rigorous question, you can mention the periodic table of the chemical elements as a background/motivation (see the faq to learn more about this).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Dmitri: 1) I asked Jacques Carette what he has in mind - he did not reply. If his suggestion was more clear, I would consider it more seriously. 2) Your reformulation of my question is not correct: I am asking about axiomatization of quantum mechanics. Formally this is not equivalent to finding eigenvalues or eigenfunctions of Schrödinger operator, since one could expect that the Mandeleev table could be proved without considering those eigenvalues. My question is absolutely clear and correct for mathematicians who have an impression of what axiomatic system is, why I should reformulate it?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Mark: there was no necessity to remove your coments here. However, I appreciated your will to talk to me, but you see, the discussion could be much more interesting, if you didn't close this topic so quickly.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: 1) I offered one possible explanation for Jacques' comment. As for 2), no, it is neither clear nor correct. For example, what do you mean by a “proof” of the periodic table? Also, a mathematical question should refer to a specific mathematical model of quantum mechanics (in our case, a specific mathematical model of an atom), e.g., you can ask a question about the Berezin-Shubin model, framed in specific terms (e.g., something about eigenvalues/eigenvectors, not about a “proof of the periodic table”).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Dmitri: 1) There is only one preson, who can explain what Jacques Carette had in mind - this is Jacques Carette himself. Instead of disputing what he wanted to say, it is much more clever to ask him this question. I did exactly this. Before he gives an answer it is senseless to refer to him. By the way, do you think it was nice of him that he dropped a hint, and disappeared immediately after that?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Dmitri: 2) I already explained what I mean by "proof". If you ask what exactly I would like to be proved, then the answer is the following. The Mendeleev table claims, for example, that the first elecctronic orbit can have only 2 electrons, the second - 8, the third - again 8, the fourth - 18, and so on. Another regularity is the structure of subshells. If quantum mechanics was an axiomatic system, all those regularities would be proved. I asked, whether this is indeed so. And it is strange for me that there are mathematicians who say that this question is unclear or incorrect.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Dmitri: 3) You compel me all the way to specify not only WHAT I would like to be proved, but also HOW I expect this to be proved. Initially you told that I should mention the Schrödinger operator, now you speak about the Berezin-Shubin model. Why is it so important? If you know what should be proved (example: the statement on the structure of orbits), why the question remains unclear until I explain how I think this must be proved (solving Schrödinger equation, or using the Berezin-Shubin model)? I used to think that the question HOW is extra in the formulation of problem.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: 1) There is no way to keep track of replies to your comment. You should contact Jacques personally if you want a reply. 2) “I already explained what I mean by "proof"”: No, you didn't. You can prove a theorem, but the periodic table is not a theorem. You have to state exactly what do you want to prove. “The Mendeleev table claims…”: You have to define in mathematical terms what do you mean by an “orbit”, “electron”, “subshell”, etc. Right now there is no mathematical content in your statements. “If you know what should be proved”: No, you don't, at least in mathematical terms.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Sergei: I think you got it right in 3), modulo terminology. “HOW” in your description refers to the fact that you must always specify the mathematical model. It is important because otherwise it is not a mathematical question. And of course I strongly disagree with your “HOW”. Even if you prove the same statement from a physical point view in two different models of quantum mechanics, form a mathematical point of view you proved two completely different statements.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Sergei: To sum up, a mathematical formulation of your question should include a specific mathematical model of quantum mechanics, mathematical definitions of all terms like “electron”, “subshell”, “orbit”, and (rigorous) mathematical statements or conjectures that you want to be proved.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Dmitri: 1) I already asked Jacques, now it's his turn. 2) That's the main point: "you must always specify the mathematical model... otherwise it is not a mathematical question". I don't agree. I can ask WHETHER THERE EXISTS A MATHEMATICAL MODEL, describing this or that phenomena, and this also will be a mathematical question for the mathematicians working in close fields (especially in fields with the same name, like "quantum mechanics"). I can also ask a mathematician WHICH THEORIES IN HIS FIELD ARE AXIOMATIC SYSTEMS, and this also will be a mathematical question. Should I explain, why?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Dmitri: In addition to what I said: when I see a text with words like "postulate" or "axiom", I can ask a specialist, whether this is an axiomatic theory or not, and whether it describes the phenomena I am interested in. Dima, I think, I understand your idea: you think that a specialist must be a robot. Moreover a badly adjusted robot, who can't understand normal language which people use for communication. Such a robot that for asking him something you should write a special program, like in FORTRAN, otherwise he will be answering only: "this is not my field, rewrite your question!".

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Dmitri: An explanation to non-robots, why this is bad: this makes impossible the communication between specialists and non-specialists (because to ask a robot anything, you must be enough sophisticated in his specific language, so you must be a specialist), this leads to the lack of understanding between different groups of people in society, and, being a kind of rudeness ("hamstvo" in Russian), this contradicts to the ethic norms. Is that clear, or I should reformulate this in some special langauge?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: 1) The point is that once you post a reply to somebody's comment, there is no way for this person to find out about this unless he looks at this page again or you inform him personally. This problem is fixed in SE 2.0 software, but MO still uses SE 1.0.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Sergei: The question “WHETHER THERE EXISTS A MATHEMATICAL MODEL describing this or that phenomenon” clearly belongs to the domain of physics, not mathematics. You other comments about robots also apply to physicists, not mathematicians. After all, we have two different sites (MathOverflow and Physics / Theoretical Physics SE) and two different sciences (mathematics and physics) for a reason, otherwise there would be just one site and one science.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Sergei: I also find your claims of rudeness completely inappropriate. Rudeness is by definition a violation of community's widely accepted norms. It is you who is violating these norms, not the MO community. Anyway, you are always welcome to start a thread on Meta about this. This comment thread is turning into a discussion, which is by itself a violation of MO rules.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
@Dmitri: You didn't comment the other two questions, which I posted: "WHICH THEORIES IN HIS FIELD ARE AXIOMATIC", and "whether this is an axiomatic theory or not...". Does this mean that you agree with me that these are mathematical questions? As to the first one - “WHETHER THERE EXISTS A MATHEMATICAL MODEL describing this or that phenomenon” - I don't agree with you, because a mathematician who work in this field should know the applications of his activity. On the contrary, a physisist very likely will say that this is not physics, since the question is about mathematical models.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Dmitri: About ethics. That's interesting: "It is you who is violating these norms". What do you mean? And I did not understand this: "you are always welcome to start a thread on Meta about this". I am not a programmer, where can I start a thread?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: I suggest that you first read the FAQ: mathoverflow.net/faq. It explains what Meta is and why discussions like this are discouraged here. It also explains what questions are appropriate here and why your question in its current form is inappropriate for MO, which should address your questions about rudeness/ethics. Also, you can see a link to Meta at the very top of any MO page. Finally, your other two questions are just as bad for MO for similar reasons, but I reiterate my request to transfer this discussion to Meta if you want to continue it.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
"I reiterate my request to transfer this discussion to Meta if you want to continue it" - does this mean that our discussion here will be deleted?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
Dima, and this statement needs proof: "It is you who is violating these norms". As well as those: "your question in its current form is inappropriate", "your other two questions are just as bad for MO for similar reasons".

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@Sergei: You are trying to continue the discussion here despite being explicitly told not to do so. I give up.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Dmitri Pavlov
Too bad I missed the party. I think this is great question! And I can't understand the reaction of some people here and on Physics.SE. Every serious mathematician, physicist, or chemist is aware of the merits of Mendeleev's table and its experimental validation. And every serious mathematician, physicist, or chemist knows similarly well that Quantum Mechanics is an idealisation. Furthermore we can not even give a closed solution for the helium atom. So I think it is a good test of QM and our mathmatical or computational skills, if we can derive the properties of M's table from SE.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Uwe Franz
Uwe, thank you for the support.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
This is a brilliant question, if I could only upvote it more than once. @Dimitri, the whole point is this is partially reverse mathematics. I.e. what axioms must we postulate in order to mathematically derive the structure of the period table which is already known experimentally to be correct and if the existing postulates of QM (appropriately formulated) are sufficient to do this. The negative reaction here is ridiculous and I also agree that you will get nonsense from physics se on this and that it will be mathematically very challenging.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Benjamin

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There is some rigorous work by Goddard and Friesecke on this, see

http://www.ma.hw.ac.uk/~chris/icms/GeomAnal/friesecke.pdf

My understanding is that even getting accurate numerics for the Schrodinger equation becomes very difficult once one has more than 10 or so electrons in play. The one regime where we do seem to have good asymptotics is when the atomic number is large but the number of electrons are small (i.e. extremely highly ionized heavy atoms).

At any rate, the foundations of the periodic table are pretty much uncontested (i.e. N-body fermionic Schrodinger equation with semi-classical Coulomb interactions as the only significant force). The main difficulty is being able to solve the resulting equations mathematically (or even numerically).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Terry Tao
answered Nov 5, 2011 by (330 points)
Terry, thank you very much for your answer, but your link is just a presentation, it does not contain even references... I would like to thank also the other people who wrote the answers. So, dear colleagues, as far as I understand, we came to a conclusion that from the point of view of logic the Mendeleev table is not explained in quantum mechanics? :)

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
References to the work given in the above slides can be found at www-m7.ma.tum.de/bin/view/Analysis/ElectronicStructure

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Terry Tao
It's strange to call "rigorous" a treatment where the charge of the nucleus is much larger then the number of electrons. In the cases considered both are around 10 - some sort of large number! Also, in chemical reactions the atoms are weakly ionized, almost neutral, so in fact $N\approx Z$. The words "use the model of non-interacting fermions" clearly hide much of the work and make a huge leap of faith from the initial equation to the answer.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Anton Fetisov
Goddard-Freiseke's work is rigorous in the regime where N is at most 10 and Z is sufficiently large; however, the predictions of that paper agree quite well with the experimental data when N and Z are both near 10, which suggests that the limitation to sufficiently large Z is a technical one rather than a fundamental one. Also, their analysis does consider Coulomb interactions between the fermions (otherwise the problem would be very easy). As I understand it, the non-interacting model is only used as a base model from which one applies rigorous perturbation theory (see p.36 of slides).

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Terry Tao
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I am not offended by the suggestion that physicists should follow the standards of mathematical proof, but I think this suggestion and the phrasing of the question demonstrate a lack of understanding of how physicists think about such things and more importantly why they put such little emphasis on axioms.

In my view it is rarely useful to think of physics as an axiomatic system, and I think this question reflects the difficulty with thinking of it as such. A different question, which is much more in tune with a physicist's point of view, would be to ask what physical description is required to explain various features of the structure of atoms as reflected in the periodic table at a prescribed level of accuracy. Until you specify what features you want to understand, and at what level of accuracy, you don't even know what the correct starting point should be. If you want just the crudest structure of the periodic table, then indeed non-relativistic quantum mechanics along with the Pauli exclusion principle will give you the rough structure as described in any standard QM textbook. If you want to understand the detailed quantum numbers of large atoms then you have to start including relativistic effects. Spin-orbit coupling is one of the most important and its effects are often summarized by a set of Hund's rules which are described in many QM textbooks or physical chemistry textbooks. If you want very accurate numerical values for ionization energies or the detailed structure of wave functions then one must do hard numerical work which probably becomes impossibly difficult for large atoms. As you ask for greater and greater precision you should eventually use a fully relativistic description. This is even harder. The Dirac equation is not sufficient, one cannot restrict to a Hilbert space with a finite number of particles in a relativistic quantum theory, and bound state problems in Quantum Field Theory are notoriously difficult. So as one asks more detailed and more precise questions, one has to keep changing the mathematical framework used to formulate the theory. Of course this process could end and there could be an axiomatic formulation of some ultimate theory of physics, but even if this were the case this would undoubtedly not be the most useful formulation for most problems of practical interest.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Jeff Harvey
answered Nov 9, 2011 by (270 points)
I think the question does not show misunderstanding of the epistemological principles under wich physicists work and develop their theories. It simply asks if it's possible to mathematically derive (in the usual mathematically rigorous sense) the Mendeleev table from Quantum Mechanics, where the latter is understood simply as a mathematical theory, not as the collection of natural phenomena it is supposed to describe.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Qfwfq
The Mendeleev table is not a mathematical theorem. It is a method of organizing atoms into groups with similar chemical properties. Quantum Mechanics is a framework which includes the nonrelativistic Schrödinger equation as well as quantum field theory. If the OP wants to know if X can be proved in a mathematically rigorous way from Y then isn't it reasonable to ask for a mathematically precise definition of both X and Y?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Jeff Harvey
@Jeff Harvey: I think it’s quite reasonable to ask without making the statements precise. Finding the right formulation of an informal idea is often as hard as proving it, or even harder. Of course, many statements are too vague to make an interesting question; but what makes this one good, I think, is that while we don’t necessarily have a precise formal statement of it in mind, “we know it when we see it”… as in Terry Tao’s answer, we do start to see it.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Peter LeFanu Lumsdaine
@Jeff Hervey: Asking questions about things which are not precisely defined is a tradition in mathematics. For example, mathematicians discussed the problems of probability theory long before Kolmogorov (in 1933) gave his axioms of probability (only after that probability got a precise definition). My question is just another example: in books on mathematical physics (e.g., in the Berezin-Shubin book) they speak very often about atoms, which as far as I understand, are not defined by now. If they can discuss atoms, why can't they discuss the Mendeleev table which describes properties of atoms?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
+ 8 like - 0 dislike

I doubt any answer will be satisfactory. My opinion is that we are still very far from a mathematical justification. If we accept the mathematical foundations of quantum mechanics, and if we make the approximation that the nucleus of the atom is just one heavy thing with $N$ positive charges, then the motion of the $N$ electrons is governed by a linear equation (Schrödinger) in ${\mathbb R}^{3N}$. The unknown is a function $\psi(r^1,\ldots,r^N,t)$ with the property (Pauli exclusion) that it has full skew-symmetry. For instance, $$\psi(r^2,r^1,\ldots,r^N,t)=-\psi(r^1,r^2,\ldots,r^N,t).$$ In practice, we look for steady states $e^{i\omega t}\phi(r^1,r^2,\ldots,r^N)$. Then $\omega$ is the energy level.

Because of the very large space dimension, one cannot perform reliable calculations on computer, when $N$ is larger than a few units. One attempt to simplify the problem has been to postulate that $\phi$ is a Slatter determinant, which means that $$\phi(r^1,r^2,\ldots,r^N)=\|a_i(r^j)\|_{1\le i,j\le N}.$$ The unknown is then an $N$-tuple of functions $a_i$ over ${\mathbb R}^3$. Of course, we do not expect that steady states be really Slater determinants; after all, the Schrödinger equation does not preserve the class of Slater determinants. Thus there is a price to pay, which is to replace the Schrödinger equation by an other one, obtained by an averaging process (Hartree--Fock model). The drawback is that the new equation is non-linear. Such approximate states have been studied by P.-L. Lions & I. Catto in the 90's.

Update. Suppose $N=2$ only. If we think to $\phi$ as a finite-dimensional object instead of an $L^2$-function, then it is nothing but a skew-symmetric matrix $A$. Approximation à la Slater consists in writing $A\sim XY^T-YX^T$, where $X$ and $Y$ are vectors. In other words, one approximate $A$ by a rank-two skew-symmetric matrix. The approximation must be in terms of the Hilbert-Schmidt norm (also named Frobenius, Schur): this norm is natural because of the requirement $\|\phi\|_{L^2}=N$. If $\pm a_1,\ldots,\pm a_m$ are the pairs of eigenvalues of $A$, with $0\le a_1\le\ldots\le a_m$, then the best Slater approximation $B$ satisfies $\|B\|^2=2a_m^2$, $\|A-B\|^2=2(a_1^2+\cdots+a_{m-1}^2)$. Not that good. Imagine how much worse it can be if $N$ is larger than $2$.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Denis Serre
answered Nov 5, 2011 by (160 points)
Even if no answer will be satisfactory, I have found the answers to this question to be very interesting...

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Tom Church
The OP asks two completely different questions: (1) "[...] if there exists a mathematical explanation of Mendeleev table in quantum mechanics? " (2) "if there is a mathematical proof of the Mendeleev table? [...] by a mathematical proof I mean a chain of logical implications from axioms of the theory [...]" This answer discusses the use of approximations, which is irrelevant to both 1 and 2. The answer to 1 is yes, and the necessity of making approximations doesn't affect the validity of the explanation. The answer to 2 is no, simply because physical theories aren't axiomatic systems.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Ben Crowell
Ben, you should explain yourself, this sounds strong: "The answer to 1 is yes". And this also: "physical theories aren't axiomatic systems". What about classical mechanics? Or probability theory?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
@SergeiAkbarov: What makes you think classical mechanics is an axiomatic system? Are you claiming that Newton's laws are an axiomatic system? I think that's clearly untrue. Or are you thinking of some restricted physical model within classical mechanics, which can be described axiomatically? That doesn't mean that classical mechanics in general is an axiomatic system. (I also don't know what you mean by the question about probability theory, since that isn't a physical theory.)

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Ben Crowell
According to V.I.Arnold, springer.com/us/book/9780387968902, Classical Mechanics can be at least considered as an axiomatic system of second order, i.e. an axiomatic system inside another axiomatic system, like General Topology, or Theory of Real numbers, or Probability Theory, etc. Hilbert's axiomatization of Euclid's geometry, en.wikipedia.org/wiki/Hilbert%27s_axioms, is another example.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
As to Probability Theory, I am sorry, this is Mathematics.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
Anyway, the words "Postulates of quantum mechanics", en.wikipedia.org/wiki/…, must mean something. If they are not the same as postulates (or axioms) in Mathematics, then there must be an explanation of their meaning. Otherwise this becomes an abuse of the trust of a listener, isn't it?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
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It depends on what you mean by proof. Even the helium atom wavefunction cannot be obtained in closed form (the way the hydrogen atom wavefunction is), so any results about the periodic table will have some level of approximation or phenomenological assumptions in them. That said, there do exists references that explain the qualitative (and quantitative) features of the periodic table based on quantum mechanics principles. Griffiths' Quantum Mechanics for instance has a very quick discussion of the periodic table around pages 186-193. It's not very complete, and also mostly not quantitative, but it nicely illustrates how quantum mechanics gives rise to the structure of the periodic table.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Physicist
answered Nov 5, 2011 by (120 points)
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I'm arriving after the war, but this is an interesting question, so I'm going to write up what I understand about it.

First of all, for a comprehensive mathematical understanding of the periodic table, you have to settle on a model. The relevant one here is quantum mechanics (for large atoms, relativistic effects start to become important, and that's a whole mess). It's entirely axiomatic, and requires no further tweaking. Then you basically have to solve an eigenvalue on a space of functions of $6N$ coordinates (ignoring spin). That gives you a "mathematical explanation" of the table, in the sense that knowledge of the solution $\psi(x_1,x_2,\dots,x_N)$ is all there is to know about the static structure of an atom. Notice that in this formulation, all electrons are tied together inside one big wavefunctions, so an "electronic state" has no meaning. Mendeleev table is not even compatible with this formulation.

Of course, solving the full eigenproblem is not possible, so all you can do is mess around with approximations. A simplistic but illuminating approximation is to completely neglect electron repulsion. Great simplification occurs, and it turns out one can speak of "electronic states". Non-trivial behaviour occurs because of the Pauli exclusion principle. This is known as the "Aufbau" principle: one builds atoms by successively adding electrons. The first electron gets itself into the lowest energy shell, then the second one gets into the same state, but with opposite spin. The third begins to fill the second shell (which has three spaces, times two because of spin), and so on. This is the basic idea behind the table, and provides a clue as to why it is organised the way it is. So this might be the theory you're looking for. It's explicitely solvable, and only requires the theory of the hydrogenoid atoms.

Of course, because of the approximations, the quantitative results are all wrong, but the organisation is still there. Except for larger elements, where the Mendeleev table is, from what I understand, an ad-hoc hack. You can improve the approximation using ideas like "screening", and this leads to the Hartree-Fock method, which still preserves the notion of shells.

Hope that helps. Then again, if you're looking for a completely logical approach to physics that'll readily explain real life, you're bound to be disappointed. Even simple theories such as the quantum mechanics of atoms are too hard to be solved exactly, which is why we have to compromise and make approximations.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Antoine Levitt
answered Nov 17, 2011 by (120 points)
Antoine, I am not against approximations. And I know that it's not necessary to solve equation for gathering information about its solutions (people in qualitative theory of differential equations demonstrate this all the way). But in any case there must be a system of axioms, definitions and propositions, otherwise it would be impossible to understand this theory. For example, Berezin and Shubin in their book speak very often about atoms, but they do not define atoms and do not make propositions about them (if not counting atoms in math. sense). Do you know a text, where this is not so?

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Sergei Akbarov
Well, as I said, you can postulate the following: An atom (or molecule, for that matter), is a core of Z protons and M neutrons described by their position, plus N electrons, each described by a wave function $\phi_i$. The electrons arrange themselve as the N lowest eigenfunctions of the eigenproblem on $H^1(R^3,C^2)$ : $-\Delta \phi_i + V(x) \phi_i = \lambda_i \phi_i$, where $V(x) = Z/|x|$ for an atom at $(0,0,0)$. Then, proposition : the eigenfunctions are organised in shells of same $\lambda_i$. The first eigenvalue has multiplicity 2, the second, multiplicity 8, etc.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Antoine Levitt
This is a toy model with little relevance for a quantitative (or even qualitative, for larger atoms) understanding of the true nature of elements. I don't think you have the correct approach, though. Physics is not a subset of mathematics, and cannot be understood as such.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Antoine Levitt
I doubt that mathematical physicists will agree with this: "Physics is not a subset of mathematics, and cannot be understood as such". But anyway if you say that everything is so simple, then there must be a reference. What you say (your definition, proposition) - where is this written?

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Sergei Akbarov
I'm actually pretty sure most mathematical physicists will agree with the sentence. Physics aims to understand the physical world, which no model alone can do without (physical, ie approximate and not rigorous) analysis. For instance, the full Schrodinger equation for an atom is mathematically perfect, and physically useless, because of its sheer complexity. So is a description of a gas by its individual molecules, which is why physicists have invented thermodynamics. But that's philosophy, and I'm not about to get dragged into that debate.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Antoine Levitt
I'm not saying that things are simple, not by a long shot, just that the theory under this particular approximation is simple. The model it's derived from is just Schrodinger's equation for an atom, whose details can be found in any quantum mechanics textbook. Assuming no electron repulsion leads to decoupling, and one can then solve according to the theory of the hydrogen atom, also standard. Any textbook will do.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Antoine Levitt
"Any textbook will do" - this sends us back to the begining of my post. Antoine, you can just try to find a book on quantum mechanics, where there is 1) a system of axioms (or postulates), 2) a phrase like "by atom we mean..." (i.e., a definition of atom), and 3) a Proposition or a Theorem about atoms. Warning: in the Berezin-Shubin book they mention atoms in Proposition 2.2 (chapter 4), but these are atoms in the sense of measure theory, they have nothing in common with physical atoms.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Sergei Akbarov
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It clearly depends upon what do you mean by the Periodic Table of elements? As usually stated, it is a vague and strictly speaking false, yet usually sufficient statement about the similarities of chemical properties of different atoms. In any case they don't repeat exactly, only with a given degree of accuracy and if you forget about some of the much more exotic behaviour, not common in reactions. If you really try to specify all of these, you'll be much better off with the common perturbation theory approach found in QM textbooks. Sure, in a sense it also defines what is being calculated, but there's also no other way to define these properties (at least I don't know any). Analysing the second-or-somewhat order of perturbation theory is a mathematically trivial, yet tedious task, but there can barely be a way to justify the order of PT rigorously, it just works. Or doesn't.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Anton Fetisov
answered Nov 13, 2011 by (40 points)
Anton, I did not understand this: "As usually stated, it is a vague and strictly speaking false". Vague and even false?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
Trivially, it's not a mathematically precise statement, so by the standards of total rigour it's vague. It's more precise (very precise) in predicting the ground state electron configurations, but less precise in predictions of chemical properties. Exceptions are rare, especially for low atomic weights, but they exits and well-known to chemists. I'm not a chemist myself, so I'll just give a few simple links: chemwiki.ucdavis.edu/Inorganic_Chemistry/… en.wikipedia.org/wiki/…

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Anton Fetisov
As it turned out (and this was not obvious for me at the beginning), what the Mendeleev table states cannot be mathematically statements at all, since the notions like atoms, electrons, shells, etc., are not precisely defined. I think those participants of this discussion, who protest against the "vague formulation of the question", should write a collective letter to a journal like "Notices of the AMS" with protest against using the words like "atom", "electron", etc., in mathematical physics, since these notions have no precise definitions.

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Sergei Akbarov
No, I don't get your sarcasm. As said before, if you're fine with the common amount of rigour in Mendeleev's table, then you should be just as fine with the common explanations in QM textbooks, so what's the point?

This post imported from StackExchange MathOverflow at 2016-03-29 20:09 (UTC), posted by SE-user Anton Fetisov
Anton, in my opinion there cannot be some amount of rigor, enough amount, common amount, etc. Either there is rigor, or there isn't, that's what I used to think about it. My aim was to understand, if there were successful attempts to interpret in the mathematical language what physicists say about what I asked. Physicists themselves can't explain this, that's why I asked this here. From what people told me I deduce that the attempts were not successful. That's enough for me.

This post imported from StackExchange MathOverflow at 2016-03-29 20:10 (UTC), posted by SE-user Sergei Akbarov

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