Music and the Making of Modern Science

Unheard Harmonies 275

the next colloquia, Bloch recalled that Schr ö dinger presented a clear account of de Bro-

glie ’ s reasoning and how it led to an explanation of Bohr and Sommerfeld ’ s quantization

rules, capable of accounting for the discrete spectral lines whose puzzling spacing we

considered in the last chapter.

When he had finished, Debye casually remarked that he thought this way of talking was rather

childish. As a student of Sommerfeld he had learned that, to deal properly with waves, one had to

have a wave equation. It sounded quite trivial and did not seem to make a great impression, but

Schr ö dinger evidently thought a bit more about the idea afterwards. Just a few weeks later he gave

another talk in the colloquium which he started by saying: “ My colleague Debye suggested that one

should have a wave equation: well, I have found one! ”^8

Debye ’ s casual intervention led Schr ö dinger to seek a wave equation for de Broglie waves.

Essentially, Debye underlined the force of the implicit analogy: what de Broglie called

waves must require some relation to the wave phenomena of mechanics, acoustics, and

optics; therefore mathematically it should be possible to express this analogy through a

wave equation of the sort known to apply in those fields. Schr ö dinger simply took the

general form of de Broglie ’ s wave and worked backward to see what partial differential

equation it could satisfy, following the pattern of sound or water waves. He found that he

was led rather directly to what now is called Schr ö dinger ’ s equation.^9

Further, once in possession of that equation, Schr ö dinger began to investigate its

general properties — even though he admitted he had no idea what was “ waving. ” Above

all, he knew that Bohr ’ s atomic theory, expressed more generally by Sommerfeld ’ s quan-

tization condition, yielded the hydrogen spectral lines, though here again neither Bohr

nor anyone else knew why these quantization conditions held, only that they “ worked ”

to explain the spectra. The furthest Bohr and Sommerfeld could go was a simple picture

in which quantization of atomic energy levels corresponded to an integral number of

“ electron waves ” fitting around the orbit so as to join back on themselves smoothly

( figure 18.2 ). But now, with a wave equation in hand, Schr ö dinger realized that these

quantized energy levels corresponded exactly with the overtone series of the waves, its

“ eigenvalues ” or “ proper values. ” As he put it at the beginning of his first seminal paper

(1926) on the hydrogen atom: “ The customary quantum conditions can be replaced by

another postulate, in which the notion of ‘ whole numbers, ’ merely as such, is not intro-

duced. Rather, when integralness does appear, it arises in the same natural way as it does

in the case of the node-numbers of a vibrating string. ”^10 Despite his aversion to music,

Schr ö dinger found himself taking the mathematical step from a wave equation to its

eigenvalues in terms of a vibrating string. Given de Broglie ’ s proposed wavelength λ for

a particle of momentum p ( λ = h/p ), Schr ö dinger was essentially taking the reverse step

of Stoney and Balmer, who went from the spectral line to the overtone: Schr ö dinger

interpreted his overtones as the stationary states that (according to Bohr) were the starting

and ending points of atomic transitions.