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Does anybody know if there exists a mathematical explanation of 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?

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migrated from theoreticalphysics.stackexchange.com Nov 7 '11 at 7:28

How do you define the Mendeleev table? i.e. what propositions do you want to see derived? –  Mitchell Porter Jun 27 '13 at 8:43
Cross-posted from mathoverflow.net/q/80146/13917 –  Qmechanic Jul 5 '13 at 17:48
What you call Mendeleev table physicist know it as the periodic table of elements ( first published by Mendeleev) en.wikipedia.org/wiki/Periodic_table . This is a table of data. Data know nothing about axioms and theoretical proofs. The job of theoretical physicists is to find a self consistent theory ( which should have axioms and proofs) which offers solutions that can fit the data. This theory exists and it is quantum mechanics. Special solutions from this theory can fit the periodic table well. When physicists say that they fit the data well, they always give an error margin. –  anna v Jul 7 '13 at 19:11
To put it another way, mathematics cannot prove that hydrogen exists, nor its energy levels. What mathematics can do is develop a theory that can give solutions that can fit the data. In physics, mathematics is a tool that is used to fit how Nature (Physis in ancient greek) behaves, allowing for predictions of new phenomena. –  anna v Jul 7 '13 at 19:17
Major edits rolled back. This is not a discussion forum, nor it is a platform for your epistemological arguments. We do science here, not pure math. Formal proofs take a back seat to correct predictions and agreement with reality. I understand that you wanted this to be on a math site, but it is on a physics site. –  dmckee Jul 7 '13 at 19:20

2 Answers 2

Yes, quantum mechanics – even non-relativistic quantum mechanics for several electrons orbiting nuclei – fully, quantitatively, and comprehensively explains all of chemistry (including biochemistry and, in fact, biology). This fact has been known since the late 1920s.

To understand the periodic character of the properties of the elements, one must realize that already the Hydrogen atom has energy eigenstates given by quantum numbers $(n,l,m)$ as well as the binary $s_z$. Energy as well as degeneracy increases as a function of $n$. When many electrons are allowed (to neutralize the positive electric charge of the nucleus), the Pauli principle (coming from the antisymmetry of the electrons' wave functions, a fact that may be deduced from quantum field theory but may be assumed as another axiom of the simplified quantum mechanical model) says that the electrons will gradually fill the states with the ever higher values of $n$. Every time one fills all states with $n<n_0$ up to some $n_0$, one gets inert gases. When one more electron is added to the new shell, we get highly reactive elements (because they include one loosely bound electron in the outer shell), and so on.

The only variation one has to add to make the calculation of the atomic energy levels exact are the electron-electron interactions (if there are at least two electrons). They slightly reorder the shells that are being filled, $1s, 2s, 2p, 3s, 3p, 4s, 3d$, and so on... The problem (aside from the basic Hydrogen problem) obviously can't be solved analytically but there exist lots of numerical techniques to find the right results and everything that has been calculated - and some of the calculations were very precise - agrees with the observations. The calculations become more complex for larger atoms (or molecules), of course. But when the size is large enough, one may use new simplifying assumptions or approximations so it's not necessary the case that it's always harder to understand/calculate larger objects.

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Dear DrSAR, the answer to your question is No, there is no joke or irony, neither intended nor unintended, in my answer. It was a well-defined question whether a subset of physics describes some class of observations and the answer is obviously Yes. I think it's good that this question was asked because people - and not only those who love to visit witches to predict the future etc. - often associate elementary low-energy physics such as atomic physics with lots of mystery that doesn't exist. All these things are almost perfectly understood. –  Luboš Motl Apr 20 '12 at 9:08
Given the answer below, it would be better to include a proof that the answer to"there is a mathematical proof of the Mendeleev table" is yes. This could be done by pointing to any evidence that this has been done. In other words, the question is not asking whether it's doable in line of principle, but whether it's done. Big difference. –  Sklivvz Dec 26 '12 at 1:18
@Sklivvz: if, as you say, it's done, then it is you who should give a reference, the OP is not obliged to divine your hints, isn't it? –  Sergei Akbarov Jul 5 '13 at 4:40
I don't understand your comment. I am not saying its done. I agree its doable in principle, but it's not done afaik. The calculations are still too complex. –  Sklivvz Jul 5 '13 at 7:41
Dear @Sklivvz, I didn't just write that it's "doable". I wrote the sketch of the actual proof. Just to remind you, the OP asked whether the periodic table of elements is explained by quantum mechanics. The answer is an unequivocal Yes. I really wrote the explanation to remind those who have thought about this important aspect of modern science. The arrangement of the elements into groups with similar properties, and which Z are in the same group with other Z, is surely explained by QM. In fact, QM has correctly calculated much more quantitative, accurate, detailed data. –  Luboš Motl Jul 9 '13 at 5:42

While quantum mechanics explains the gross features of the periodic system, many fine details of the periodic table of elements are computable numerically from various approximations to QED, but are conceptually ill understood. See, e.g.,

Eric R. Scerri, How Good Is the Quantum Mechanical Explanation of the Periodic System? Journal of Chemical Education 75 (1998), 1384.

Scerri also wrote a book on the subject (The periodic table: its story and its significance, 2007). Several book reviews are available online:
Werner Kutzelnigg writes in his review: ''I am personally skeptical whether a genuine PS [periodic system] of the elements that incorporates all chemical properties of the elements (e.g., their tendency to form covalent, ionic, semipolar, multicenter, or hypervalent bonds) will ever be formulated. Another issue about which one would like to learn more is whether the periodic system has a chance to survive in the realm of superheavy elements.''
Michael Laing describes (for the Platinum Metals review) in his review anomalies of platinum.

It is difficult to derive from the periodic table (or from quantum mechanics) precise, generally valid laws about chemical elements. In a 2008 paper for the Americal Scientist, The past and future of the periodic table, Scerri writes about the predictive power of the periodic system, ''if one considers all of Mendeleev's many predictions of new elements, his powers of prophecy appear somewhat less impressive, even to the point of being a little worrying. In all Mendeleev predicted a total of 18 elements, of which only nine were subsequently isolated. [...] the Davy medal, which predates the Nobel Prize as the highest accolade in chemistry, was jointly awarded to Mendeleev and Julius Lothar Meyer, his leading competitor, who did not make any predictions. Indeed, there is not even a mention of Mendeleev's predictions in the published speech that accompanied the joint award of the Davy prize. It therefore seems that this prize was awarded for the manner in which the two chemists has successfully accommodated the then-known elements into their respective periodic systems rather than for any foretelling.''
''it is possible to predict that subsequent main shells of the atom can contain a maximum of 2, 8, 18 or 32 electrons. This is in perfect agreement with the lengths of periods in the chemist's periodic table. The simple quantum mechanical theory does not, however, account for the repetition of all period lengths except for the first one. Indeed, this problem has continued to elude theoretical physicists until quite recently. Appropriately enough, it was a Russian physicist, the late Valentin Ostrovsky, who recently published a theory to explain this feature, although it is not yet generally accepted. Although the theory is too mathematically complicated to explain here, Ostrovsky's work and some other competing accounts demonstrate that the periodic table continues to be an area of active research by physicists as well as chemists even though it has existed for nearly 140 years.''

For a very recent review on the expert level, see the paper The physics behind chemistry, and the Periodic Table by Pyykkü. He mentions that a number of important effects (such as the color of gold, the liquidity of mercury, or the voltage of a lead-acid battery) need QED (more precisely the Dirac-Coulomb-Breit approximation to QED rather than the textbook nonrelativistic Schroedinger equation) for their correct explanation. He treats the periodic system shorter than the title would suggest, but makes up for this in this paper.

Of interest may also be papers by Bonchev and Kibler; the latter relates the periodic system to the dynamical symmetry group $SO(4,2)$ of the hydrogen atom.

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Why knock on Mendeleev? He had the right idea, and he was heckled enough in his own time. He was many decades ahead of all his competitors, and all his sound predictions were verified, and those that weren't were only because of isotope anomalies that he couldn't have understood back then. You might as well criticize Gell-Mann for not figuring out everything about quarks in 1964. –  Ron Maimon Apr 30 '12 at 5:07
@RonMaimon: Stating an interesting fact about his prediction doesn't make his accomplishemnts smaller. Your comment makes sense with the definition of ''sound'' = ''what agreed with reality'', but how would you have told what was sound before the experiments that decided upon agreement with reality? –  Arnold Neumaier Apr 30 '12 at 11:35
This comment is a little naive--- the experimental data sufficient to establish the table was known in the 19th century, as demonstrated by the fact that Mendeleev did it. The further experiments served to seal the coffin on alternate theories, and were political necessities, not logical necessities. The previous experiments were sufficient for the logical development, and the resistance to Mendeleev was ignorant and reactionary, as is the resistance to all new ideas at all times. –  Ron Maimon Apr 30 '12 at 15:25
Then why did Mendeleev propose the 9 unsound predictions if it was all known to him? With 9 correct out of 9 predictions he would have been a much better predictor than with 9 correct out of 18, which reveals a lack of good prediction rules based on what he already knew. –  Arnold Neumaier Apr 30 '12 at 16:34

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