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§ Phase i · The Quantum Crisis · Chapter 03

Balmer's ladder

A Swiss schoolteacher with no laboratory of his own found a pattern in the rainbow that hydrogen emits when heated, a single formula that fit four numbers perfectly. Nobody understood why for thirty years.

In the autumn of 1885, a sixty-year-old mathematics teacher in Basel sat down with a sheet of paper and four numbers. The numbers were wavelengths, measured in a laboratory he had never set foot in, of the four bright bands of light that hydrogen gives off when you run an electric current through it. They were old numbers by then, refined by Anders Ångström a decade earlier to a precision of one part in ten thousand: 656.21, 486.07, 434.01, and 410.12 nanometres. Red, blue-green, blue, violet. A spectroscopic chord, played in the world’s simplest atom.

Up to that evening the four numbers had been a curiosity. Physicists had been collecting them, and lines from sodium and iron and a hundred other elements, for thirty years, ever since

Robert Bunsen and Gustav Kirchhoff had pointed a prism at the flame of their newly invented burner in Heidelberg and discovered that every element vapourised into the flame painted its own signature in colour. Sodium gave a yellow pair of lines so sharp you could measure them like a ruler. Potassium glowed violet. Iron sprayed a bewildering forest of hundreds of lines across the spectrum. Each element had a fingerprint. Nobody knew why.

Kirchhoff had even formulated the deepest law of all in 1859, a principle so general that it survived every subsequent revolution in physics intact: a substance absorbs at exactly the wavelengths it emits. The black bands threading through the spectrum of the Sun, the so-called Fraunhofer lines catalogued in 1814 by a Bavarian optician working with the world’s best prisms, corresponded one-to-one with the bright bands that hot earthly elements gave off in the lab. The Sun’s atmosphere was, in some sense, a backlit fingerprint of its composition. Spectroscopy was on its way to becoming a science of remote chemistry. By the 1870s, astronomers were arguing about whether stars had the same elements as Earth based on whether their dark lines matched terrestrial bright lines. (Helium, famously, was discovered in the Sun before it was discovered on Earth, by the wavelength of a yellow line nobody could match to any known element.)

But amid this cataloguing fever, hydrogen sat apart. It was the simplest element, the lightest in the new periodic table that Mendeleev had laid out in 1869, and its spectrum was a kind of insult to theory. Only four lines in the visible range. Four. Iron had hundreds; calcium had dozens; even sodium had its famous doublet. Hydrogen, the element you would expect to be simplest, gave you the smallest puzzle and the most provocative one. The four wavelengths sat in spectroscopy tables looking like a numerical poem nobody could scan.

What Balmer noticed that autumn, and he was not a physicist by training (he had spent his career teaching geometry to schoolgirls at a finishing school), was that the four wavelengths of hydrogen were not random. They obeyed a single equation. Plug in m = 3 and the formula spat out 656 nm; plug in m = 4 and it gave 486 nm; m = 5, m = 6, every line in the visible spectrum, to within Ångström’s measurement error. The equation had one constant. He called it h (a coincidence of notation; Planck’s h was still fifteen years in the future), with a value of about 364.6 nm. We will follow modern convention and call it B, for Basel.

How Balmer arrived at the formula has the flavour of a parable about pure mathematics. His friend Eduard Hagenbach-Bischoff, the physics professor at the University of Basel, had handed him the four wavelengths as a numerical curiosity, knowing Balmer’s love of patterns. Balmer had spent his life looking for hidden ratios in everything from the proportions of the Egyptian pyramids to the architecture of Solomon’s temple, a recreation he pursued in a religious-mystical key, the way some of his contemporaries kept songbird collections. He attacked the four hydrogen wavelengths the same way he would have attacked any numerical puzzle: by trying to express them as a single function of a small integer.

The breakthrough came when he noticed that each wavelength was very nearly B times a fraction of the form m² ⁄ (m² − 4), with m taking the values 3, 4, 5, and 6 in turn. The 4 in the denominator looked arbitrary until you saw it as 2². The integer 2 was lurking in the formula like a secret signature. Balmer himself had no theory of what the integer 2 meant (that would have to wait for Bohr), but he was confident enough in his arithmetic to predict, in the same 1885 paper, the wavelength of a fifth line at m = 7. It had not been measured. He wrote down a value of 397.0 nm. Ångström’s successors found it the following year, exactly where Balmer had said it would be, deep in the violet near the edge of the visible range. The four lines became five. Then six.

The paper Balmer submitted to the Annalen der Physik in 1885 (it had already appeared earlier in the proceedings of the Naturforschende Gesellschaft in Basel, a quieter venue) is two and a half pages long. It contains one figure, no derivations, and exactly one formula. The tone is modest, almost apologetic. Balmer notes that he hopes the formula may “be of some use to physicists.” It was also, as it turned out, the last physics paper he would ever write. He returned to geometry, to his architectural studies of Solomon’s temple, and to his daughters, of whom he had six. He died in 1898, two years after Henri Becquerel discovered radioactivity, without having seen his formula explained.

To understand why Balmer’s formula was such a bombshell, you have to picture how strange spectral lines looked to a nineteenth-century physicist. The dominant theory of light was Maxwell’s: light was a continuous electromagnetic wave, and a heated body should radiate a smooth, continuous spread of wavelengths. Black-body radiation, the subject of our first chapter, did roughly that; a hot iron poker glows over a broad band of colour. But an atom of hydrogen, excited by an electric spark, did not. It emitted only four colours. Four discrete spikes, with absolute silence in between.

It was as if a piano had been built that could play only four notes. And not four notes you could pick at random: four notes related to each other by a hidden arithmetic. Nobody at the time could say what was vibrating, or why it would prefer some frequencies over others. The atom itself was still hypothetical in 1885. Ludwig Boltzmann was being heckled at conferences for believing atoms existed at all, and the electron was twelve years away from being discovered by J. J. Thomson at the Cavendish Laboratory in 1897. Hydrogen was simply “the lightest gas.” It had a number (atomic weight 1), a chemical behaviour, and a spectrum, and the spectrum had no business being so musical.

The closest analogy a physicist of the 1880s could grasp was a mechanical resonator. A violin string emits a fundamental and an overtone series: the harmonics, integer multiples of the fundamental frequency. You might therefore expect an atom, if it had some internal mechanical structure, to vibrate at a fundamental and its harmonics: f, 2f, 3f, 4f. Hydrogen does not do this. The Balmer wavelengths, converted to frequencies, do not form a harmonic series. They form a converging series, crowding closer together as you climb the ladder, accumulating toward a limit at 364.6 nm in the near ultraviolet. Beyond that limit the lines pile up into an apparent continuum. This is not how strings ring. It is not how cavities resonate. It is not how any classical oscillator behaves. The pattern in Balmer’s formula belonged to no mechanical model anyone could imagine.

Three years after Balmer’s paper, in 1888, a Swedish physicist named Johannes Rydberg, who had been struggling with the spectra of the alkali metals and looking for a unifying empirical law, looked at the Basel formula and saw something Balmer had missed. He rewrote it in terms of the reciprocal of wavelength, what spectroscopists called the wavenumber (measured in inverse centimetres), and the equation suddenly opened up. Instead of one constant B and an awkward fraction, Rydberg’s version had a single universal constant (now called R, the Rydberg constant) and a difference of two reciprocal squares. The formula became:

1/λ = R · (1/n₁² − 1/n₂²)

Balmer’s series was the special case where n₁ = 2. Rydberg’s leap was to suggest that other series existed for n₁ = 1, n₁ = 3, n₁ = 4, and so on: entire families of spectral lines, most of them lurking in the ultraviolet and infrared, invisible to the eye. He had not seen them. He simply trusted the arithmetic. There is something thrilling about this: Rydberg had taken a small empirical observation about one element’s visible spectrum and turned it into a prediction machine. Pure pattern-recognition, pushed past the data, betting that nature would keep playing the same tune in registers humans could not yet hear.

The Balmer series is characterized by the electron transitioning from n ≥ 3 to n = 2, where n refers to the radial quantum number or principal quantum number of the electron. The transitions are named sequentially by Greek letter: n = 3 to n = 2 is called H-α, 4 to 2 is H-β, 5 to 2 is H-γ, and 6 to 2 is H-δ. As the first spectral lines associated with this series are located in the visible part of the electromagnetic spectrum, these lines are historically…

Rydberg was right. Over the next forty years, every single one of those predicted series was found. Theodore Lyman, working in the ultraviolet at Harvard between 1906 and 1914, discovered the n₁ = 1 series, the Lyman series, whose first line lies at 121.6 nm, deep in the far ultraviolet. Friedrich Paschen, working with bolometers in Tübingen in 1908, discovered the n₁ = 3 series in the near infrared. Frederick Sumner Brackett, at Johns Hopkins, found the n₁ = 4 series in 1922. August Herman Pfund found n₁ = 5 in 1924. Curtis Humphreys, working in the far infrared at the National Bureau of Standards, completed the next rung (n₁ = 6) as late as 1953. Each new series fell exactly where Rydberg’s formula said it would, sometimes to six significant figures. The hydrogen atom turned out to be radiating in an infinite ladder of discrete frequencies, and a single equation, with one fitted constant, captured all of it.

The fit was too good. That is not a phrase you usually hear in physics. Theories are praised when they predict; formulas are praised when they fit. But the Rydberg formula was disturbing precisely because it had no underlying theory at all. There was no F = ma behind it, no Maxwell equations, no thermodynamic argument. Heinrich Kayser, the German spectroscopist who compiled the great Handbuch der Spectroscopie between 1900 and 1934, called the formula “a wonderful gift from the spectroscope to the theorist, who has so far refused to receive it.” The theorists had not refused. They had no idea what to do with it.

And yet, even with every series found and every line predicted, the formula was a stranger to the rest of physics. It worked. Nobody knew why. The numbers n₁ and n₂ were integers, not parameters fitted to data, but pure whole numbers from the counting integers. Why should an atom care about integers? The Rydberg constant R had no derivation; it was simply a fitting parameter, an empirical knob. The formula didn’t connect to mechanics, didn’t connect to Maxwell’s equations, didn’t connect to thermodynamics. It was a riddle written in the language of arithmetic.

Look at the structure for a moment. The Rydberg formula tells you that the wavenumber of any line in hydrogen is the difference of two quantities, each labelled by an integer. The first quantity is R / n₁². The second is R / n₂². The spectral line itself is what you see when you subtract them. The natural reading (though no nineteenth-century physicist allowed himself to say it out loud) is that R / n₁² and R / n₂² are real quantities the atom possesses, and the spectral line is a kind of transaction between them. The atom has a ledger of allowed values; emitting a photon means moving from one row of the ledger to another, and the wavelength of the photon equals the difference. Walther Ritz, a brilliant young Swiss theorist who died of tuberculosis at thirty-one, formalised this guess in 1908 as the combination principle: every spectral line of an element is the difference of two members of a discrete set of “terms” characteristic of that element. He could prove the principle from the spectra. He could not prove why the terms existed.

For the last fifteen years of the nineteenth century and the first thirteen years of the twentieth, Balmer’s formula sat on the table of physics like a sealed envelope. Everyone could read what was written on the outside. Nobody could open it. Henri Poincaré, in his lectures on mathematical physics, called the spectra of elements “the most difficult problem in physics,” and added that the atom, whatever it was, “must have a structure of impenetrable simplicity.” J. J. Thomson, who had discovered the electron in 1897 and who had built a plum-pudding model of the atom around it, could not derive Balmer’s formula from his model and never publicly admitted it bothered him. Lord Rayleigh, the most distinguished spectroscopist of his generation, mentioned the Balmer series in his Cambridge lectures with the resigned formulation: “This curious result … must mean something, but it is not yet known what.”

The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The classification of the series by the Rydberg formula was important in the development of quantum mechanics. The spectral series are important in astronomical spectroscopy for detecting the…

↘ Derive: from Balmer's B to Rydberg's R

Balmer’s original equation, in his 1885 notation, was:

λ = B · m² / (m² − 4)

where m = 3, 4, 5, 6 ... indexes the four visible lines and B ≈ 364.6 nm is the Balmer limit, the wavelength the series converges to as m → ∞. The factor of 4 in the denominator is 2², hinting at a hidden integer that Balmer himself never named.

Rydberg saw the structure. Take reciprocals:

1/λ = (m² − 4) / (B · m²) = (1/B) · (1 − 4/m²) = (4/B) · (1/4 − 1/m²)

Now substitute 4/B = R (the Rydberg constant) and 2² = 4:

1/λ = R · (1/2² − 1/m²)

Generalising the lower level from 2 to any positive integer n₁ < m:

1/λ = R · (1/n₁² − 1/n₂²)

This is the full Rydberg formula. Balmer’s series is n₁ = 2. Lyman’s series is n₁ = 1. Paschen’s series is n₁ = 3. The constant R is universal across hydrogen (and, with a small correction for nuclear mass, across every hydrogen-like ion). Numerically, R = 4/B = 4 / (364.6 nm) ≈ 1.0974 × 10⁷ m⁻¹.

The puzzle hiding inside this algebra is that n₁ and n₂ are integers. Empirically forced. No classical model of an atom (a planet of charges orbiting a sun of charges) gives any reason for integers to appear. That is the riddle Bohr inherits in the next chapter.

The reason Balmer’s formula matters to our story is not the formula itself. It is what the formula was telling everyone, in arithmetic so clean a schoolteacher could find it: that the atom is a quantised thing. That its inner life is not continuous. That something inside the hydrogen atom, whatever it was, could exist only in certain discrete states, and could emit light only when it jumped between them. Each spectral line was a transition. Each integer was an address. The atom was a building with floors, not a hillside; the photon was the elevator chime as you crossed between floors; the floor numbers, finally, were the integers Balmer had stared at for thirty years.

Balmer never said any of this. He had no model of the atom; he had a formula. Rydberg never said it either; he had a generalisation. Even the great spectroscopists who spent forty years confirming the Lyman, Paschen, and Brackett series did not say it. The integers were sitting in the equation, in plain view, daring someone to notice. The puzzle was being delivered to physics in three pieces from three directions at once. Planck had delivered the quantum of action, h, in 1900, almost as a bookkeeping trick to fix the black-body curve. Einstein had delivered the photon in 1905, claiming that light itself came in discrete bundles of size hν. And Balmer had delivered, twenty years before either of them, the empirical proof that hidden inside the simplest atom there was a discrete, integer-labelled structure that emitted light only at certain specific frequencies.

Read together, the three pieces formed a sentence. Planck’s h set the size of the bundle. Einstein’s E = hν related the bundle to a frequency. Balmer’s 1/λ = R(1/n₁² − 1/n₂²) provided the frequencies the bundle could carry. Substitute Planck’s relation into Balmer’s formula and you get something extraordinary: the energy of each emitted photon equals hc · R · (1/n₁² − 1/n₂²), which factors as the difference of two energies of the form −hcR/n². The atom must possess discrete energy levels of that form. The integers n are the labels on the levels. Photons are emitted when an electron drops from one level to another, with an energy precisely equal to the difference.

None of the three men who supplied the pieces (Planck, Einstein, Balmer) drew that conclusion themselves. The synthesis required a fourth man, willing to mix two heresies (Planck’s quanta, Einstein’s photons) with a third (the breakdown of classical mechanics at atomic scales), and to do so in writing, with a model concrete enough to derive Balmer’s formula from first principles. The man who finally did it was a twenty-seven-year-old Danish theoretical physicist named Niels Bohr, working as a junior research assistant in Manchester in the spring of 1913, in the middle of a three-part paper that everyone (including the physicists of his own generation) thought was a little mad.

§

Balmer’s four numbers had been sitting on the table for twenty-eight years when Bohr finally turned them upside down. He kept Planck’s quantum, kept Einstein’s photon, and proposed that the integers in Balmer’s formula were not curiosities but coordinates, labels for the rungs of a ladder inside the atom itself. To climb the ladder, he had to break every rule of classical mechanics.

next chapter → Bohr's leap