We shall attempt to derive the Stefan-Boltzmann Law
\[ \Phi = \sigma T^4 \]
where \(\Phi\) is the radiant energy flux and \(\sigma\) is a constant with dimension \(\text{watts}/{\text{cm}}^3\cdot{\text K}^4\), but where we do not care about the value of the constant. To begin, consider the number of modes of a vibrating string clamped at both ends that have frequency \(\nu' \le \nu\). The relation between frequency and mode nuuber is
\[ \nu = \frac{ck}{2\pi}, \]
that is, frequency is proportional to wave number. Therefore the number of modes less than give frequency is bounded by a quantity proportional to the frequency:
\[ N(\nu) \propto k \]
Consider next a large cubical box. A standing wave is characterized by mode numbers \(k_i\) and frequencies \(\nu_i\). The same reasoning shows that the number of modes is counted by
\[ N(\nu) \propto k^3 \]
It follows (by differentiaton) that the density of states \(n(\nu)\) is bounded by a constant times \(k^2\).
Planck’s Law for energy density:
\[ u(\omega,T) = \frac{h\omega^3}{\pi^2 c^3}\frac{1}{e^{h\omega/T} - 1} \]
Now integrate over all angular frequencies
\[ u(T) = \int_0^\infty u(\omega,T)d\omega = \frac{h}{\pi^2 c^3}\int_0^\infty \frac{\omega^3}{e^{h\omega/T} - 1} d\omega \]
Introduce the substituion \(x = h\omega/kT\). Then
\[ \begin{aligned} u(T) &= \frac{h}{\pi^2 c^3} \frac{k^4 T^4}{h^4}\int_0^\infty \frac{dx}{e^x - 1} dx \\ &= a T^4 \end{aligned} \]
for a constant
\[ a = \frac{k^4}{\pi^2 c^3}\int_0^\infty \frac{x^3 dx}{e^x - 1} . \]
which depends only on the speed of light and Boltzmann’s constant, plus the dimensionless matheamtical quantities \(\pi\) and the integral
\[ \int_0^\infty \frac{dx}{e^x - 1} \]
The integral on the right is standard and goes by many names: it is a particular case of a Bose–Einstein integral, the polylogarithm, or the Riemann zeta function \(\zeta(s)\). The value of the integral is
\[ \Gamma(4)\zeta(4) = \frac{\pi^4}{15} \]
Structural / scaling viewpoint
Abstracting the physics:
- The mode space is 3‑dimensional, so the density of states per unit \(\nu\) is homogeneous of degree 2 in \(\nu\).
- The microscopic quantum of energy is linear in the thermal occupation is a dimensionless function of the ratio .
- So any equilibrium spectral energy density with these properties must have the structure
\[ u(\nu,T) = \nu^3 F\left( \frac{h\omega}{kT} \right) \]
for some dimensionless \(F(x)\), which Planck’s calculation identifies as a constant times \(1/(e^x - 1)\). That is the mathematical reason the intensity (per unit frequency) carries a cubic power of frequency: one power from the photon energy, two from the 3D density of modes.