Planck's Law - Derivation

Derivation

See also: Gas in a box

Consider a cube of side L with conducting walls filled with electromagnetic radiation in thermal equilibrium at temperature T. If there is a small hole in one of the walls, the radiation emitted from the hole will be characteristic of a perfect black body. We will first calculate the spectral energy density within the cavity and then determine the spectral radiance of the emitted radiation.

At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The three wavelengths λ₁, λ₂, and λ₃, in the three directions orthogonal to the walls can be:

where the n_i are integers. For each set of integers n_i there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:

The quantum number r can be interpreted as the number of photons in the mode. The two modes for each set of n_i correspond to the two polarization states of the photon which has a spin of 1. Note that for r = 0 the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at absolute temperature T.

According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by:

Here

The denominator Z(β), is the partition function of a single mode and makes P_r properly normalized:

Here we have implicitly defined

which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function:

This formula, apart from the first vacuum energy term, is a special case of the general formula for particles obeying Bose–Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero.

If we measure the energy relative to the ground state, the total energy in the box follows by summing over all allowed single photon states. This can be done exactly in the thermodynamic limit as L approaches infinity. In this limit, ε becomes continuous and we can then integrate over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between ε and ε + dε as g(ε)dε, where g(ε) is the density of states (which we'll evaluate in a moment), then we can write:

To calculate the density of states we rewrite equation (1) as follows:

where n is the norm of the vector n = (n₁, n₂, n₃):

For every vector n with integer components larger than or equal to zero, there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of dε corresponds to shell of thickness dn = (2L/hc)dε in n-space. Because the components of n have to be positive, this shell spans an octant of a sphere. The number of photon states g(ε)dε, in an energy range dε, is thus given by:

Inserting this in Eq. (2) gives:

From this equation one easily derives the spectral energy density as a function of frequency and as a function of wavelength u_λ(T):

where:

And:

where

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form using only elementary functions. Substituting

in Eq. (3), makes the integration variable dimensionless giving:

where J is a Bose–Einstein integral given by:

The total electromagnetic energy inside the box is thus given by:

where V = L3 is the volume of the box.

This is not the Stefan–Boltzmann law (which provides the total energy radiated by a black body per unit surface area per unit time), but it can be written more compactly using the Stefan–Boltzmann constant σ, giving

The constant 4σ/c is sometimes called the radiation constant.

Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral radiance of radiation exiting the small hole is

which yields

It can be converted to an expression for B_λ(T) in wavelength units by substituting by c/λ and evaluating

Note that dimensional analysis shows that the unit of steradians, shown in the denominator of left hand side of the equation above, is generated in and carried through the derivation but does not appear in any of the dimensions for any element on the left-hand-side of the equation.

This derivation is based on Brehm & Mullin 1989.