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§ Phase ii · Matter Waves · Chapter 02

Schrödinger's winter

In the last days of 1925, a 38-year-old professor left Zurich for an alpine villa with a married woman whose name he never recorded, two pearls in his ears to block out noise, and a paper by Louis de Broglie in his luggage. He came back with the equation that would write every atom, every molecule, and every chemical bond into mathematics. The four papers he then published in 1926 are the founding document of wave mechanics. He spent the rest of his life uneasy about what they meant.

By the autumn of 1925, the physics of the atom was a half-built bridge with traffic already on it. Heisenberg had announced his matrix mechanics in July, an austere, abstract scheme of non-commuting tables of numbers that gave the right answers without ever drawing a picture. Bohr’s old quantized orbits had been quietly retired. Nobody had any idea what an electron actually was, and the senior generation, raised on Maxwell’s fields and Lagrangian mechanics, found the new abstraction insufferable. Einstein had called Heisenberg’s algebra a “true witches’ multiplication table” and confessed that he could not bring himself to believe it. There had to be a wave-mechanical alternative, something continuous, something one could visualise. Into that gap walked an Austrian professor in his late thirties named Erwin Schrödinger.

Schrödinger was, by 1925, a slightly improbable revolutionary. He was a careful, classically-trained theorist at the University of Zurich, fluent in four languages, a published poet, an enthusiastic amateur botanist, and a confirmed romantic in the most literal sense. He read Schopenhauer for pleasure. He kept three or four affairs running in parallel and recorded each of them in a private diary in a code his wife Anny tolerated for reasons her own. His scientific work to that point had been respectable and varied (statistical mechanics, color theory, a bit of general relativity) but nobody, in the autumn of 1925, would have predicted that he was about to put his name on the most consequential equation of the twentieth century. He was on the shortlist for promotions, not for genius.

What changed everything was a thesis. In Paris, Louis de Broglie had defended a doctoral dissertation in 1924 arguing that if light, long known as a wave, could behave as a particle (Einstein had insisted on this since 1905), then by symmetry, particles long known as matter ought to behave as waves. Every electron carried a wavelength λ = h/p, inversely proportional to its momentum. The committee had not known what to make of it. Paul Langevin sent the thesis to Einstein and asked, more or less, “is this nonsense?” Einstein read it and replied that it was “a first feeble ray of light on this worst of our physics enigmas.” The thesis was approved. And quietly, in a letter to Schrödinger in late 1925, Einstein praised de Broglie’s idea and suggested that it deserved further investigation.

Schrödinger gave a colloquium on de Broglie’s thesis at the University of Zurich in November 1925. In the audience was Peter Debye, the head of the physics institute and a senior colleague. After the talk, Debye delivered what would become one of the most generative compliments in the history of science. He said, in effect: this is all very nice, but if matter is a wave, then it must obey a wave equation. Where is yours? Schrödinger, embarrassed, admitted that he did not have one. He set out, over the following weeks, to find it.

This is where the ski trip comes in. Around December 23, 1925, Schrödinger left Zurich for the Villa Herwig in Arosa, a small ski resort in the Swiss Alps. With him went a woman who has been variously described in the secondary literature as “an old Viennese girlfriend” and “a mistress,” but whose name has never been disclosed and whose identity, to this day, remains unknown. (Anny Schrödinger remained at home in Zurich; she was aware of the trip; her marriage operated, by mutual consent, on terms that contemporaries found puzzling and that biographers have learned to describe gently.) Schrödinger took with him de Broglie’s thesis, Einstein’s encouraging letter, the proofs of Hermann Weyl’s textbook on differential equations, his pearls, and a few notebooks. He stayed for roughly two and a half weeks. When he returned in mid-January 1926, he had something in hand that nobody had had before.

What did he actually do, mathematically? The plan was straightforward once stated. De Broglie had given matter a wavelength. Standing waves on a string have discrete frequencies (the harmonics) because the string is pinned at both ends; the boundary conditions select which wavelengths are allowed. The hydrogen atom is, in effect, a three-dimensional box. If the electron is a wave, then the allowed energies of hydrogen ought to come out as the discrete eigenvalues of some wave equation in the Coulomb potential, the way the harmonics of a violin string come out as the discrete eigenvalues of a string equation. The mystery of why Bohr’s orbits were quantized would dissolve. The quantization would not be a postulate. It would be a fact about boundary conditions.

The equation Schrödinger wrote down (after several false starts, including a relativistic version that gave wrong answers for fine structure and was set aside, only to be rediscovered later by Klein and Gordon) is the one we now teach. In its time-dependent form for a single non-relativistic particle of mass m moving in a potential V(x, t), it reads

i ℏ ∂ψ/∂t = - (ℏ² / 2m) ∇²ψ + V(x, t) ψ

where ψ(x, t) is a complex-valued function on space and time called the wavefunction. The right-hand side is the Hamiltonian operator Ĥ acting on ψ. The kinetic-energy term is the Laplacian, ∇², which counts how curved the wavefunction is at each point. The potential V is whatever well or bump or atomic Coulomb attraction the particle finds itself in. The whole equation is, in plain English, a recipe: tell me the energy landscape (V) and the present shape of the wave (ψ now), and I will tell you the shape of the wave a moment later.

In four foundational papers published through 1926, collectively titled “Quantisierung als Eigenwertproblem” (Quantization as an Eigenvalue Problem) I, II, III, and IV, Schrödinger laid out the equation, solved it for the most important cases (the free particle, the harmonic oscillator, the rigid rotor, and the hydrogen atom), demonstrated that it reproduced every quantitative prediction of Bohr’s older model without any of Bohr’s postulates, and showed that his wave mechanics and Heisenberg’s matrix mechanics, though they looked nothing alike, were mathematically equivalent. It was the most productive twelve months of theoretical physics anyone had ever had. The hydrogen calculation in particular was a triumph. Allow the wavefunction to decay smoothly at infinity (no piling up of probability at large distances, since the electron must be somewhere) and require it to be single-valued (no contradictions in physical interpretation), and the only allowed energies fall out as

E_n = - (13.6 eV) / n²,    n = 1, 2, 3, ...

exactly the Bohr-Balmer ladder, with the same integer n, only now derived from a smooth wave equation in three dimensions rather than postulated from a circular-orbit picture nobody could justify. The quantum number n labelled the radial nodes of a wavefunction, not a literal orbit. The angular momentum quantum number ℓ labelled the angular nodes. And the magnetic quantum number m labelled the orientation. Three integers, three boundary conditions, one Coulomb well, and chemistry, the whole periodic table in embryo, came out clean.

The mathematics of the hydrogen calculation came from a corner of nineteenth-century analysis few physicists had ever met: the special functions named after Laguerre, Legendre, and the spherical harmonics tabulated by Laplace. Schrödinger leaned heavily on his Zurich colleague Hermann Weyl, one of the very best mathematicians of the era, to handle the boundary conditions cleanly. Weyl, in his memoirs, described Schrödinger in this period as working “in a late erotic outburst of his life,” a phrase that has since been quoted in every popular account of the equation and that contains, beneath the wry euphemism, an entirely accurate observation. The Arosa companion (whoever she was) had given Schrödinger the physical and emotional space to think. The equation came out of a long, quiet, isolated stretch of attention. The rest is implementation.

Ĥ φ(x) = E φ(x)

- (ℏ² / 2m) (1/r²) d/dr (r² dR/dr) - (e² / (4 π ε₀ r)) R = E R.

a₀ = 4 π ε₀ ℏ² / (m e²) ≈ 5.29 × 10⁻¹¹ m,    E = - e² / (8 π ε₀ a₀) ≈ - 13.6 eV.

The reception was electric. Einstein wrote to Schrödinger that the work was “the product of a true genius.” Planck, then in his late sixties, called it “epoch-making.” Within months, every theoretical physicist who could understand a partial differential equation was applying the new wave mechanics to a different problem: the helium atom, the harmonic oscillator with a perturbation, the tunneling of an alpha particle out of a uranium nucleus. The hydrogen molecule fell in 1927 to Heitler and London. The covalent bond, the central object of all chemistry, became a calculation. Schrödinger himself shared the 1933 Nobel Prize in Physics with Paul Dirac, “for the discovery of new productive forms of atomic theory.” He delivered the lecture, took the gold medal, and went back to brooding.

For Schrödinger had a problem with what he had unleashed. He had imagined, in 1926, that the wavefunction ψ was a real physical entity, a smeared-out charge distribution that just happened to obey his equation. The electron was, he wanted to believe, an oscillating cloud. Within months, Max Born had shown that this could not be right. The wavefunction was, instead, a probability amplitude: its squared modulus |ψ|² gave the probability of finding the particle at a point, but ψ itself was not a thing in space. We will read Born’s paper in detail in the next chapter. Schrödinger never accepted Born’s interpretation, just as he never accepted Heisenberg’s matrices, just as he never accepted Bohr’s complementarity. By the early 1930s he had moved to Berlin, then fled the Nazis to Oxford and eventually to Dublin, and he watched his own equation get fitted into a probabilistic, indeterminate worldview he found repellent. In a 1952 paper looking back on the discoveries of his youth, he wrote one of the most quoted sentences in the philosophy of physics: “I don’t like it, and I’m sorry I ever had anything to do with it.”

He did not, of course, mean the equation. The equation works. Every prediction it has ever made has been confirmed to the limit of measurement, often to a dozen or more decimal places. The structure of every atom in the periodic table, the geometry of every molecule, the operation of every semiconductor, every laser, every MRI machine, every quantum computer in development today, all rest on the same first-order partial differential equation Schrödinger wrote down at the Villa Herwig over the New Year of 1926. What he could not bring himself to like was the interpretation: the irreducibly probabilistic, observer-dependent picture of nature that Bohr and Born had wrapped around his mathematics. The man who lit the lamp spent the rest of his life squinting at the shadows.