A Suggested Parsing of Planck’s Spectral Distribution Formula

G.R.Dixon, 12/24/2006

Planck’s spectral distribution for black body radiation is

. (1)

(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/l^{4}. 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:

. (2)

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.

Figure 1

r(l,T)

Figure 2

N(l,T)

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. As l 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.