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We'll start out with the **wave equation** in n dimensions

The **solutions of the wave equation** are

This is easily proven, as the second time derivative of *u* yields

In real space, the solutions *u* are harmonic waves with the frequency *ω*, travelling along the direction of with the wavelength

In reciprocal space (k-space), these solutions represent only points (i.e. states) with the position .

Given a n-dimensional box in real space with a sidelength of *L* and a volume of

Example of a twodimensional k-space. Drawn in are several states (black points), the first Brilloin zone of one state (blue rectangle) and the volume increment d*V _{(k)}* (red annulus circular ring).

The numbers of states having a wave number between *k* and *k*+d*k* are given by the numbers of possible polarisations *p* times the volume increment d*V _{(k)}* in k-space divided by the space one state occupies, i.e.

This leads immediatly to the density of states, which is defined as the numbers of states divided by the volume of our n-dimensional box in real space and wave number increment d*k*.

Of course, the density of states can also be expressed with respect to the frequency *ω*, the wavelength *λ* or the energy *E*:

The distribution of the internal energy *u(λ)* is easily obtained by multiplying the energy with the density of states and the probability that a photon will have the specified energy. The last factor is known as the Bose-Einstein distribution.

The internal energy *U* of our n-dimensional box can be determined by integrating it's distribution *u(λ)* from zero to infinity and multiplying it with the volume of the box *V*.

If you insert the energy distribution and try to evaluate the expression for the internal energy, following definite integral will most likely appear:

The other thermodynamic properties (specific heat *c _{V}*, entropy

For the derivation of the black body radiation, we have to imagine a n-dimensional hemisperical container, with a constant internal energy distribution *u(λ)* inside and a small hole d*A* cut into the center of it's face. The vector pointing out of this small hole is .

The intensity distribution of the black body radiation is the amount of energy flow *I(λ)* coming out of this hole. The energy flow from the solid angle element d*Ω* can be expressed as the energy d*E(&lambda, &Omega)* passing the hole in the time interval between *t* and *t* + d*t*, divided by the length of the interval d*t*.

Let's analyse the volume d*V*, which gets passed through the hole in the time interval described above. Because the photons are moving isotropic inside our container, only the fraction d*Ω* divided by the total solid angle *Ω _{R}*, of the photons inside the volume d

When expressing the volume d*V* by the vector product of a length and the area vector and plugging this into the formula for the energy flow, we find the photon's velocity vector

While the magnitude of the velocity vector must be the speed of light *c*, it's direction can be expressed by the unitary vector pointing into the direction *Ω*. Altogether, this yields for the energy flow from the solid angle element d*Ω*

Integrating over the solid angle *Ω _{R/2}* of our hemisphere, we obtain for the intensity distribution of the black body radiation

The Wien's Law is easily calculated for a given radiation distribution. It expresses the wavelength of the intensity maximum *λ _{max}* as a function of the temperature

Also the Stefan-Boltzmann Law can be derived quite easily from the radiation distribution. It is just the integral of the distribution *I(λ)* from wavlength *λ* = 0 to infinity.

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