Edmond Laguerre

Edmond Nicolas Laguerre was born on April 9, 1834 in Bar-le-Duc, a quiet town in northeastern France perched above the river Ornain. He was the bookish son of a provincial bourgeois household, weak enough in health that schoolmasters routinely worried whether he would survive the year. The same fragility would shadow him for the rest of his life and ultimately kill him at fifty-two.

He entered the École Polytechnique in 1853, then as now the most demanding scientific school in France. The school marched its students through a brutal curriculum of analysis, geometry, and mechanics under instructors who treated mathematics the way a drill sergeant treats a parade ground. Laguerre passed through without distinction in the rankings but with a clear gift for pure analysis. He left in 1855 with a commission as an artillery officer, the standard exit for Polytechniciens of his rank, and spent the next nine years in garrison towns doing mathematics in his spare hours.

The artillery posting was not a detour. Nineteenth-century artillery officers were expected to know more pure mathematics than most university lecturers. Laguerre wrote his first paper at twenty, on the projective geometry of conics, and continued publishing throughout his army years. By 1864 his work was strong enough that Polytechnique recalled him from the artillery to serve as a répétiteur, a senior tutor drilling cadets through the material the professors had laid out. He held that post for almost twenty years, rising in 1883 to a chair at the Collège de France in mathematical physics. The Académie des Sciences elected him a member the same year. He had reached the inner circle of French mathematics, but his lungs had reached their limit.

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The shape of Laguerre’s work was unusual. He scattered short, dense papers across three subjects: projective geometry, the theory of equations, and a new branch of analysis concerned with orthogonal polynomials. His complete works fill two volumes and contain more than 140 entries, almost none of them long. He thought in compact, finished propositions.

The result that would outlive everything else came from a memoir he submitted to the Journal de mathématiques pures et appliquées in 1879. It studied a family of polynomials orthogonal under the weight e^{-x} on the half-line from zero to infinity. He defined them, derived their recurrence, proved their orthogonality, and used them in the theory of continued fractions. The polynomials had been glimpsed before by Abel and others, but Laguerre was the first to lay out their full apparatus. In modern notation a Laguerre polynomial Lₙ(x) is a polynomial of degree n such that the integral from zero to infinity of Lₙ(x) Lₘ(x) e^{-x} dx vanishes whenever n is different from m. The exponential weight is what makes the family special: it tames their growth at large x and gives them the right shape to describe anything that decays exponentially far from the origin.

He had no way to know what his polynomials were good for. The atom had not been discovered, the electron had not been discovered, and the notion that a particle might be a wave was not yet a serious idea. Laguerre’s universe was Maxwell’s: continuous fields, continuous matter, planets and projectiles obeying classical differential equations. He worked out the polynomials because the structure was there.

The tuberculosis that had threatened him since boyhood caught up with him in his late forties. He had married in 1862 and raised two daughters in the Quartier Latin. In the spring of 1886 he took the train south to Bar-le-Duc to convalesce at home. He died there on August 14, 1886, fifty-two years old, leaving behind two volumes of Œuvres that his colleague Charles Hermite assembled a decade later.

Fifty years passed.

In 1926 a Viennese physicist named Erwin Schrödinger sat down to solve a wave equation he had written that winter for the hydrogen atom. The equation separated, in spherical coordinates, into an angular part (the spherical harmonics of Laplace) and a radial part. The radial equation, after the change of variables Schrödinger chose, turned out to be a generalized Laguerre equation. Its solutions, the radial wavefunctions of the bound electron, were the associated Laguerre polynomials multiplied by a decaying exponential. Every shell and subshell from hydrogen to uranium is described, in the central-field approximation, by some piece of Laguerre’s 1879 machinery. The exponential weight he had picked for purely analytical reasons turned out to be the natural language of an electron’s quantum reach away from the nucleus.

Laguerre’s name is now on the polynomials, on a differential equation, on a quadrature rule, on a group of geometric transformations, and on a crater of the Moon. The polynomials are by far the most consequential. Open the equation for a 3d radial function and you will find, buried under the constants, an associated Laguerre polynomial of degree two; for a 4f state, one of degree three. Schrödinger neither named nor invented these objects. He inherited them from a French artillery officer who had died forty years earlier still believing his work belonged to pure analysis. Laguerre stands for the moment, recurring throughout this book, when a piece of mathematics worked out for its own sake turns out to be exactly the right language for an atom that had not yet been imagined.

He is the radial half of every orbital we will draw.