266 Chapter 17
musical antecedents. To sample the distribution of energies inside the oven, Planck intro-
duced an imaginary resonator as a probe. This resonator he assumed to be an idealized
pendulum or harmonic oscillator, and thus an idealization of a vibrating string. This theo-
retical contrivance allowed him to express the relation between the radiant energy in the
cavity of the oven at a certain frequency (chosen to be the resonant frequency of the
imaginary resonator itself) and the entropy of that ambient radiation. Then Planck could
apply the second law of thermodynamics to the entropy by calculating the number of pos-
sible ways in which the available energy might be distributed over a whole ensemble of
imaginary resonators, each probing the oven ’ s radiation at its own particular resonant
frequency.
In the previous chapter, we saw the prehistory of this hypothesis in Maxwell ’ s assump-
tion that molecules could be treated as vibrating bodies whose modes of resonant oscilla-
tion are then visible as the spectrum of the substance formed by those molecules. Under
that assumption, the walls of the cavity itself would be formed of an immense number of
those vibrating molecules. Thinking of their vibrations in terms of sound, the oven (or
indeed any chunk of matter) is essentially a chorus, an assemblage of many individual
“ singers, ” each sounding its own particular note or resonant frequency. The light inside
the heated oven, thus, would be like the chord formed by all those individual voices,
sounding a continuum of pitches, according to classical theory.
Planck at times calls these “ Hertzian oscillators, ” referring to Heinrich Hertz ’ s 1886
experiments that used a spark gap and capacitor to induce a rapidly oscillating electric
spark capable of producing a detectable amount of electromagnetic radiation, as predicted
by Maxwell ’ s equations ( figure 17.5a ). To produce the rapid interruption of the electric
spark, Hertz used a vibrating contact that, when operating, generates an audible pitch ( ♪
sound example 17.6); his electromagnetic vibrator is also in fact a source of ordinary sound
vibrations. Indeed, the physical structure of these mechanical vibrators is essentially identi-
cal to the reed of a harmonium ( figure 17.5b ), though Hertz used a metal vibrator that
could conduct the electricity for the spark. Thus, Hertz ’ s physical contrivance has a con-
tinuous lineage with the other imaginary and real oscillators we have already discussed,
including those envisaged by Stoney and Balmer. As they had used a vibrator to model
light-emitting atoms and molecules, Planck uses his oscillator to model an idealized probe
of the ambient radiation emitted by the excited atoms in the oven cavity.
Far more often than “ oscillator, ” Planck uses the term “ resonator, ” the term Helmholtz
also used for his instrument tuned to resonate sympathetically at a certain pitch (see figure
14.6). Planck ’ s resonator is likewise coupled with the vibrations it both registers and
retransmits.^30 As Helmholtz used many resonators to map out the frequency spectrum, each
tuned to a pitch of interest given by a harmonium, Planck too envisages a whole ensemble
of resonators, each capable in principle of being tuned to any given pitch.^31 In that sense,
his thought experiment envisages an imaginary harmonium, an instrument capable of