Unheard Harmonies 277
The emission frequencies [of the hydrogen spectral lines] appear therefore as deep “ difference tones ”
of the proper vibrations themselves. It is quite conceivable that on the transition of energy from one
to another of the normal vibrations, something — I mean the light wave — with a frequency allied to
each frequency difference , should make its appearance. One only needs to imagine that the light
wave is causally related to the beats , which necessarily arise at each point of space during the transi-
tion; and that the frequency of the light is defined by the number of times per second the intensity
maximum of the beat-process repeats itself.^12This extraordinary passage, with its tumultuous language and jumpy italics, expresses
Schr ö dinger ’ s struggle to yoke together the aspects of light and sound that provide the
mathematical analogies he used to describe the atomic events underlying spectral emission.
Above all, he seems forced to these verbal contortions because he is trying to picture
processes that resolutely defy any visualization. His allusions to sound emerge under the
pressure of trying to visualize the unvisualizable: in the atomic realm, the extended analogy
with sound helped him interpret his equation, lacking any other means to connect it to
visual reality.
To be sure, after his initial insight in terms of sound vibrations, Schr ö dinger ’ s visual
orientation and even his color theory came into play when he considered how an atomic
field of force might affect the resulting “ overtones ” ; in that case, the difference between
relative values of potential and kinetic energies acts to “ curve ” the manifold in which
Schr ö dinger ’ s waves act, which one might compare to the way that the human visual
system “ curves ” the manifold of color perception. Thus, Schr ö dinger ’ s equation represents
a kind of synthesis that embeds overtones within a “ curved, ” non-Euclidean environment
dependent on the energy of the system and the forces at work there. In that sense, his
equation combines the idiomatically musical element of overtones with the visual compo-
nent of rays traversing a curved manifold ( figure 18.3 ). Using this generalized wave
description, Schr ö dinger was able to put forward an extended analogy: the new quantum
mechanics is to ordinary mechanics as diffractive wave optics is to the geometrical optics
of rays. Yet despite his free use of optical or visual metaphor, Schr ö dinger frankly acknowl-
edged he had no idea what these “ waves ” might be, only that his wave equation plus the
boundary conditions typical for vibration problems led to overtones corresponding to the
observed hydrogen spectra. Nor was it clear in what “ space ” his wave equation operated;
at first, he hoped the waves (whatever they were) were physical in the sense of occupying
ordinary space, but when he began to consider more complex atoms with N electrons,
he realized that instead the wave equation would have to be formulated within a 3 N -
dimensional “ configuration space. ”
So great was the power of the generalized mathematics of the wave equation that, even
without any concept of the nature of the waves involved, Schr ö dinger could deduce their
overtones and experimental implications. Though proud of his optical analogies,
Schr ö dinger was left mainly with negative conclusions about the intuitive or visual meaning
of his wave equation: “ no special meaning is to be attached to the electronic path itself, ”