14/07/2021

Marcelo Viana in Folha: the mathematics of the periodic table

Crédito: International Science Concil

Reproduction of Marcelo Viana's column in Folha de S.Paulo

The periodic table contains two elements in the first row, eight in the second and third, 18 in the fourth and fifth, and 32 in the sixth and seventh. I pointed out last week that 2, 8, 18, and 32 are double the perfect squares 1, 4, 9, and 16. Where do these numbers come from? To explain, let's start with a more familiar problem.

When we pluck a loose string, it can move in many different ways. But if the ends are fixed, as in a guitar, for example, there are far fewer possible movements, to the point that they can be listed.

There's the basic movement, with a single "belly" vibrating up and down. Two bellies that move alternately in opposite directions, with the midpoint of the string stationary. Three alternating bellies, separated by two stationary points. And so on (the notes of a real guitar are combinations of these movements).

Oscillatory phenomena, such as the motion of a string, are described by the wave equation, formulated by Jean D'Alembert (1717–1783) in 1746. The Sturm–Liouville theory, initiated by Jacques Sturm (1803–1855) and Joseph Liouville (1809–1882) in the 1830s, and which continues to be researched today, shows that solutions subject to boundary conditions (for example, that the ends are fixed) can be listed using integers.

In quantum models of the atom, the movement of electrons around the nucleus is described by another equation, the Schrödinger equation, but much of the mathematical theory still holds true. And the fact that the orbit of each electron must "close" when it completes a revolution around the nucleus is a boundary condition.

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