Planck's Law from Bosonic Photon Gas

1. The Photon Gas: A Gas of Non-Conserved Bosons

The problem of blackbody radiation concerns the spectral distribution of electromagnetic radiation in thermal equilibrium with the walls of a cavity at a fixed temperature $T$. According to the quantum theory of radiation, this electromagnetic field is composed of discrete energy quanta called photons. To apply statistical mechanics, we must first understand the key properties of this "photon gas."

The following video provides an excellent introduction to this concept. It clearly explains why photons are treated as bosons and, most importantly, establishes the critical property that distinguishes a photon gas from an ordinary gas of atoms.

The conclusion from the video is central to our entire derivation. For a gas of non-conserved particles like photons, the chemical potential $\mu =0$. Recalling the Bose-Einstein distribution for the average occupation number of a state with energy $ϵ$:

Setting $\mu =0$ and using the photon energy relation $ϵ=\hslash \omega$, we find the average number of photons in a single electromagnetic mode of frequency $\omega$:

This is precisely the expression for the average excitation number of a quantum harmonic oscillator that we found in the last lesson. This confirms our model: the thermal energy of the electromagnetic field can be understood by summing the energies of its constituent modes, each behaving as a quantum harmonic oscillator populated by a bosonic quasi-particle, the photon.

2. Counting the Modes: The Density of States

To find the total energy in the cavity, we must sum the average energy of every possible electromagnetic mode. Since the frequencies of these modes form a near continuum, we need to replace a direct sum with an integral. This requires knowing how many modes exist within a given frequency interval, a quantity known as the density of states, $g(\omega )$.

The derivation involves considering the allowed wavevectors in a finite box and then taking the thermodynamic limit. David Tong's lecture notes provide a clear and mathematically rigorous derivation of this quantity.

From this reading, the number of electromagnetic modes in a volume $V$ with angular frequency between $\omega$ and $\omega +d\omega$ is:

This expression tells us that the number of available modes grows quadratically with frequency. This is a crucial piece of the puzzle.

3. Assembling the Pieces: The Planck Distribution

We now have all the necessary components to derive the Planck distribution. The energy density of radiation, $u(\omega ,T)$, within a small frequency interval $d\omega$ is the product of three factors:

1. The number of modes per unit volume in that interval: $\frac{g(\omega )}{V}d\omega =\frac{{\omega }^2}{{\pi }^2{c}^3}d\omega$.
2. The average number of photons occupying each of these modes: $⟨n_{\omega }⟩=\frac{1}{e^{\beta \hslash \omega }-1}$.
3. The energy carried by a single photon in that mode: $\hslash \omega$.

The video you watched earlier walks through this exact calculation, combining the density of states with the Bose-Einstein distribution.

Multiplying the three terms gives the energy per unit volume in the frequency range from $\omega$ to $\omega +d\omega$:

Rearranging terms, we obtain the Planck Radiation Law:

This formula gives the spectral energy density of blackbody radiation as a function of angular frequency $\omega$ and temperature $T$.

This result was a triumph of quantum statistical mechanics.

• At low frequencies ($\hslash \omega \ll k_{B}T$), we can approximate $e^{\hslash \omega /k_{B}T}\approx 1+\hslash \omega /k_{B}T$. The formula reduces to $u(\omega ,T)\approx \frac{{\omega }^2{k}_{B}T}{{\pi }^2{c}^3}$, which is the classical Rayleigh-Jeans law. This law agrees with experiment at low frequencies but diverges at high frequencies—the ultraviolet catastrophe.
• At high frequencies ($\hslash \omega \gg k_{B}T$), the exponential term dominates, and $u(\omega ,T)\propto {\omega }^3{e}^{-\hslash \omega /k_{B}T}$. This exponential suppression prevents the divergence and correctly describes the experimental observation that the spectrum falls to zero at high frequencies.

For completeness, the derivation can also be performed more formally by first calculating the log of the grand partition function for the photon gas and then finding the internal energy, as shown in the KIT lecture notes.

Conclusion

In this lesson, we successfully applied the principles of quantum statistical mechanics to derive the Planck distribution for blackbody radiation. By modeling the electromagnetic field as a collection of harmonic oscillators and treating the excitations—photons—as a gas of non-conserved bosons, we arrived at a formula that perfectly matched experimental data and resolved a major crisis in classical physics.

Key Takeaways:

• A gas of photons is a system of non-interacting bosons whose particle number is not conserved, leading to a chemical potential of $\mu =0$.
• The average number of photons in a mode with frequency $\omega$ is given by the Bose-Einstein distribution $⟨n_{\omega }⟩=(e^{\beta \hslash \omega }-1{)}^{-1}$.
• The density of electromagnetic modes in a volume $V$ is $g(\omega )d\omega =V\frac{{\omega }^2}{{\pi }^2{c}^3}d\omega$, including a factor of 2 for polarization.
• Combining these elements gives the Planck distribution for spectral energy density: $u(\omega ,T)=\frac{\hslash }{{\pi }^2{c}^3}\frac{{\omega }^3}{e^{\hslash \omega /k_{B}T}-1}$.

We have now seen Bose-Einstein statistics applied to both phonons (in the harmonic oscillator model) and photons. This concludes our module on the formalism of quantum statistical mechanics. In the next module, we will shift our focus to the other family of quantum particles—fermions—and begin by exploring the properties of the ideal Fermi gas, a model of fundamental importance for understanding electrons in metals and the structure of white dwarf stars.