A Suggested Parsing of Planck’s Spectral Distribution Formula
Planck’s spectral distribution for black body radiation is
(Given a unit volume cavity, Eq. 1 also states the energy in the cavity.) Rayleigh and Jeans had earlier geometrically deduced that the possible number of standing waves is 8p/l4. And Einstein later suggested that radiant energy, when present, is quantized in amounts hc/l. What Planck appears to have done is throttle back the number of possible standing waves to a number (say N) of standing waves that actually contain energy:
According to this view, each (unit volume of) standing wave then contains a single photon of energy hc/l.
Fig. 1 plots Planck’s r(l,T) at a temperature of 1500 Kelvin, and Fig. 2 plots N(l,T). Note the similar shapes.
It is rather interesting how these two curves complement one another. At very small wavelengths the individual photons energies are high, but few accessible waves are actually populated. Asl increases the photon energies decrease but the number of occupied waves more than compensates. Finally, at lT=b (where b is Wien’s displacement constant), r(l,T) peaks. Thereafter the decreasing photon energies and the attenuation of N collectively ensure that little energy is found in the longer wavelengths.