![]() The elements in the same column have similar chemical properties.įor example, the noble gases at the far right are all pretty stable on their This question prompts me to ask something more specific about the periodic table.Īs far as I know, the main significance of the periodic table is that The questions I ask here are hopefully much easier, but still research-level Mathematical problem motivated by atomic structure, tackled by people like Fefferman and Lieb. Was regarded as an example of an important Prove the stability of matter, based only on the Schroedinger equation Mathematical world, maybe I am unaware of how much things have changed since I was a Since I don't keep too well in touch with cultural trends in the $\epsilon=1$? But maybe this is just as difficult as giving a full account of the structures using analysis.īy the way, I certainly wouldn't like to complain, but it is a bit puzzling to me why some people regard this Is there a way to see the numbers we see occurring via the spectral flow of this family as we go from $\epsilon =0 $ to Moving around a nucleus of charge $Z$ looks like As mentioned by Jeff and Antoine, the Hamiltonian for a system of $N$ electrons I like Neil's formulation, but I think I am asking something much more naive.Īllow me to start by providing more background for mathematicians whose knowledge is as hazyĪs mine. If we don't want to go against this convention, we need to stick to the versionĪfter receiving the nice answers from Jeff and Antoine, I realized that I should sharpen the question I just wanted to point out that my question `what is a shell?' has a clear-cut answer in this usage. If this is confusing to you, I suggest you don't worry about it. For example, in the fourth row of the periodic table, starting with potassium (K) and up to copper (Cu), they say both the $M$-shell and the $N$-shell are incomplete. Rather, the conventional description of the phenomena says that more than one shell can be incomplete in an atom. (In case you're wondering, the representations after $V_3\otimes S=f$ are labeled consecutively in the alphabet.) The key point is that, in this form, a shell does not consist of a grouping of subshells of similar energies when more than a few electrons are present. O-shell: $5s\oplus 5p \oplus 5d\oplus 5f\oplus 5g $Īnd so on. N-shell: $4s\oplus 4p \oplus 4d\oplus 4f $ As far as I can tell, the shells are simply the direct sums of the following form: It doesn't help much with my questions, but it does describe the convention regarding the term 'shell'. Secondly, I finally read the wikipedia article on shells. The definitive term for any given occurrence of an irreducible representation in $L^2\otimes S$ seems to be subshell. That is, the individual wavefunction is an orbital, while the representation itself might be referred to as the so-and-so orbitals, in the plural. So the eigenfunctions in the second occurrence of the representation $V_1\otimes S$ will be called the $3p$-orbitals. I will do so here, and leave the text below as written, in case the temporary confusion is helpful to other interested non-experts.įirstly, as far as I can tell, the word 'orbital' does seem to refer to a wavefunction, not a representation. Having read some more here and there, I should make some terminological corrections, in case I mislead anyone with my ignorance. Or to look up a proper reference like Atkin's book on physical chemistry. ![]() ![]() That is more or less independent of experimentsĬonfirmed by difficult computations using approximation schemes.Īnyone interested in the background and details is invited to read the incoherent paragraphs below, So I believe the essence of whatĪ natural mathematical explanation for the stability Such an alternative definitionĭoes not seem to exist, and it is not at all obvious how to come up with The wikipedia article on noble gases offers an amusing formulation whereby, forĪ noble gas, 'the outer shell of valence electrons is considered to be "full"' (my emphasis).Īll this led to my initial confusion: I thought that the term 'shell' must mean somethingĮlse for multi-electron systems in a manner adapted to the answer (A). The s and p subshells (corresponding to the representations $V_0\otimes S$ and $V_1\otimes S$)įilled in the outermost shell, and this is not a full shell once the atom is bigger than Argon. If we take the usual definition of a shell, all the noble gases but Helium have just The question is thereby somewhat narrower in scope than my original query, but I would be happy to have this focused version answered.įrequently heard is definitely wrong. So I thought I would summarize it in the simplest form I could manage and add it to the title. I've succeeded in making the question entirely unintelligible with all my additions. ![]()
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