Count thermal photons with zero chemical potential

Each electromagnetic mode is a harmonic oscillator with energy spacing hbar times angular frequency. Its thermal state permits occupation zero, one, two, and so on. Because photon number can adjust through equilibrium interaction with the walls, no independent conserved-number multiplier is imposed: the photon chemical potential is zero. Summing the geometric occupation weights gives a Bose occupation. This applies to equilibrium blackbody light; driven lasers and approximately number-conserving photon arrangements can require a different effective description.

The thermal excitation energy per mode is its photon energy times mean occupation. A temperature-independent zero-point term is omitted from this thermal spectrum; questions involving changes in vacuum energy or boundary stresses need their own treatment. At low photon energy compared with thermal energy the occupation becomes large and the mode approaches the classical equipartition value. At high energy the exponential suppresses occupation. This high-frequency suppression prevents the ultraviolet divergence obtained by assigning thermal energy to infinitely many classical modes.

In a large vacuum cavity, shell counting in wavevector space gives mode density proportional to angular frequency squared. There are two independent transverse polarizations. Counting them once produces the density used here. Multiplying mode density by energy per photon and Bose occupation gives thermal spectral energy density per angular-frequency interval. Density per ordinary frequency differs because angular frequency is two pi times ordinary frequency; the interval Jacobian must accompany any change of variable.

Substitute hbar times angular frequency divided by thermal energy into the total-energy integral. Every frequency factor and the integration interval contributes a temperature factor, yielding total energy density proportional to temperature to the fourth power. The remaining dimensionless integral is pi to the fourth divided by fifteen. A spectral maximum depends on whether density is expressed per frequency or per wavelength, so a peak cannot be converted between those plots simply by inverting its horizontal coordinate.

Photons carry momentum equal to energy divided by c. In an isotropic cavity, the average squared directional cosine is one third, so radiation pressure is energy density divided by three. A small hole emits only outward-going photons; weighting their crossing rate by the directional cosine gives flux equal to c times energy density divided by four. The one-third and one-quarter factors describe different angular averages and cannot be interchanged.

The escaping energy flux is the Stefan–Boltzmann exitance of an ideal black surface at that temperature. A real object's emissivity, spectral response, surrounding radiation, and geometry alter its net heat exchange. A single directed beam does not obey the isotropic pressure formula. The quadratic lab can verify the mode-density frequency-square factor alone. Its parabola has neither Bose occupation nor a thermal spectrum, so the Planck curve and total temperature-fourth-power law remain analytical deductions.