Planck Radiation Law

Consider electromagnetic radiation inside a box whose walls are held at the constant temperature $T$. We know that electromagnetic radiation of angular frequency $\omega$ is quantized into photons whose energy is $\epsilon = \hbar\,\omega$. (See Section 3.3.8 and 4.1.2.) Thus, given that photons are indivisible, the allowed energy levels of such radiation are equally spaced, with spacing $\hbar\,\omega$. In this respect, each frequency state acts like a harmonic oscillator of angular frequency $\omega$. (See Section 4.3.7.) According to Equation (5.390), the mean energy of a harmonic oscillator of angular frequency $\omega$ that is in thermal equilibrium with a heat reservoir of temperature $T$ is

$$\langle\epsilon\rangle= \frac{\hbar\,\omega}{\exp(\hbar\,\omega/k_B\,T)-1}.$$ (5.467)

Here, we have neglected the zero-point energy, $(1/2)\,\hbar\,\omega$, in Equation (5.390) because there is no electromagnetic zero-point energy.

Let $u(\omega)$ be the electromagnetic energy per unit volume associated with electromagnetic waves whose angular frequencies lie between $\omega$ and $\omega+d\omega$. It follows that

$$V\,u(\omega)\,d\omega = \rho(\omega)\,\langle \epsilon\rangle \,d\omega,$$ (5.468)

where $\rho(\omega)\,d\omega$ is the number of electromagnetic wave states whose angular frequencies lie between $\omega$ and $\omega+d\omega$. Making use of Equations (5.457) and (5.467), we deduce that

$$u(\omega) = \frac{\hbar\,\omega^3}{\pi^2\,c^3\,[\exp(\hbar\,\omega/k_B\,T)-1]}.$$ (5.469)

This result is known as the Planck radiation law, after Max Planck who first obtained it in 1900.

Figure: 5.7 The Planck radiation law. Here, $\omega_0=k_B\,T_0/\hbar$ and $u_0=\hbar\,\omega_0^{\,3}/(\pi^{2}\,c^{3})$, where $T_0$ is an arbitrary scale temperature. The dashed, solid, and dash-dotted curves show $u/u_0$ for $T/T_0=0.75$, $1.0$, and $1.25$, respectively. The dotted curve shows the locus of the peak emission frequency.

Consider the classical limit $\hbar\rightarrow 0$. In this limit, the previous expression becomes

$$u(\omega)= \frac{k_B\,T\,\omega^2}{\pi^2\,c^3}.$$ (5.470)

This result is known as the Rayleigh-Jeans radiation law, after Lord Rayleigh and James Jeans who derived it in the first decade of the twentieth century. The Rayleigh-Jeans law is equivalent to the assumption that each electromagnetic wave state possesses the classical energy $k_B\,T$ predicted by the equipartition theorem. (See Section 5.5.5.) The total classical energy density of electromagnetic radiation is given by

$$u_{\rm tot} = \int_0^{\infty} u(\omega) \,d\omega = \frac{k_B\,T}{\pi^{2} \,c^{3}} \int_0^{\infty} \omega^{2}\,d\omega.$$ (5.471)

This is an integral that obviously does not converge. Thus, according to classical physics, the total energy density of electromagnetic radiation inside an enclosed cavity is infinite. This is clearly an absurd result. In fact, this prediction is known as the ultra-violet catastrophe, because the Rayleigh-Jeans law usually starts to diverge badly from experimental observations (by over-estimating the amount of radiation) in the ultra-violet region of the spectrum.

The Planck radiation law approximates to the classical Rayleigh-Jeans law for $\hbar\,\omega \ll k_B\,T$, peaks at about $\hbar\,\omega\simeq 3\, k_B\,T$, and falls off exponentially for $\hbar\,\omega \gg k_B\,T$. See Figure 5.7. The exponential fall-off at high frequencies ensures that the total energy density of electromagnetic radiation inside an enclosed cavity remains finite. The reason for the fall-off that it is very difficult for a thermal fluctuation to create a photon with an energy greatly in excess of $k_B\,T$, because $k_B\,T$ is the characteristic energy associated with such fluctuations.