It isn’t surprising that math is involved in the periodic table. Decades before the existence of atoms was shown for sure (Einstein in 1905 on Brownian motion — https://physicsworld.com/a/einsteins-random-walk/) Mendeleev arranged the known elements in a table according to their chemical properties. Math is great at studying and describing structure, and the periodic table is full of it.

What *is* surprising, is how periodic table structure arises from math that ostensibly has absolutely nothing to do with chemistry. Here are 3 examples.

The first occurred exactly 60 years ago to the month in grad school. The instructor was taking a class of budding chemists through the solution of the Schrodinger equation for the hydrogen atom.

Recursion relations are no stranger to the differential equations course, where you learn to (tediously) find them for a polynomial series solution for the differential equation at hand. I never really understood them, but I could use them (like far too much math that I took back then).

So it wasn’t a shock when the QM instructor back then got to them in the course of solving the hydrogen atom (with it’s radially symmetric potential). First the equation had to be expressed in spherical coordinates (r, theta and phi) which made the Laplacian look rather fierce. Then the equation was split into 3, each involving one of r, theta or phi. The easiest to solve was the one involving phi which involved only a complex exponential. But periodic nature of the solution made the magnetic quantum number fall out. Pretty good, but nothing earthshaking.

Recursion relations made their appearance with the solution of the radial and the theta equations. So it was plug and chug time with series solutions and recursion relations so things wouldn’t blow up (or as Dr. Gouterman put it, the electron has to be somewhere, so the wavefunction must be zero at infinity). MEGO (My Eyes Glazed Over) until all of a sudden there were the main quantum number (n) and the azimuthal quantum number (l) coming directly out of the recursions.

When I first realized what was going on, it really hit me. I can still see the room and the people in it (just as people can remember exactly where they were and what they were doing when they heard about 9/11 or (for the oldsters among you) when Kennedy was shot — I was cutting a physiology class in med school). The realization that what I had considered mathematical diddle, in some way was giving us the quantum numbers and the periodic table, and the shape of orbitals, was a glimpse of incredible and unseen power. For me it was like seeing the face of God.

The second and third examples occurred this year as I was going through Tony Zee’s book “Group Theory in a Nutshell for Physicists”

The second example occurs with the rotation group in 3 dimensions, which is a 3 x 3 invertible matrix, such that multiplying it by its transpose gives the identity, and such that is determinant is +1. It is called SO(3)

Then he tensors 2 rotation matrices together to get a 9 x 9 matrix. Zee than looks for the irreducible matrices of which it is composed and finds that there is a 3×3, a 1×1 and a 5×5. The 5×5 matrix is both traceless and symmetric. Note that 5 = 2(2) + 1. If you tensor 3 of them together you get (among other things 3(2) + 1) = 7; a 7 x 7 matrix.

If you’re a chemist this is beginning to look like the famous 2 L + 1 formula for the number of the number of magnetic quantum numbers given an orbital quantum number of L. The application of a magnetic field to an atom causes the orbital momentum L to split in 2L + 1 magnetic eigenvalues. And you get this from the dimension of a particular irreducible representation from a group. Incredible. How did abstract math know this.

The third example also occurs a bit farther along in Zee’s book, starting with the basis vectors (Jx, Jy, Jz) of the Lie algebra of the rotation group SO(3). These are then combined to form J+ and J-, which raise and lower the eigenvalues of Jz. A fairly long way from chemistry you might think.

All state vectors in quantum mechanics have absolute value +1 in Hilbert space, this means the eigenvectors must be normalized to one using complex constants. Simply by assuming that the number of eigenvalues is finite, there must be a highest one (call it j) . This leads to a recursion relation for the normalization constants, and you wind up with the fact that they are all complex integers. You get the simple equation s = 2j where s is a positive integer. The 2j + 1 formula arises again, but that isn’t what is so marvelous.

j doesn’t have to be an integer. It could be 1/2, purely by the math. The 1/2 gives 2 (1/2) + 1 e.g two numbers. These turn out to be the spin quantum numbers for the electron. Something completely out of left field, and yet purely mathematical in origin. It wasn’t introduced until 1924 by Pauli — long after the math had been worked out.

Incredible.