Calculating the energy density: Our black body is a perfect cube. You see one lower edge of it. On every plane, a wave has to have the value of zero (a knot - the black dots). This boundary condition leads to the (classical allowed) energy density

How said, the energy is defined by the radiation inside the black body. The behaviour of this radition is descrived by Maxwell’s euqations.

A solution of these equations are plane waves. The differential equation for it is

. A solution is where c is the speed of light, the frequency and the wave vector (Direction of movement).

In a closed black body, most likely a cube with the side length of L, a wave has to have a knot: This means it’s value at the boundaries has to be zero (The black dots in the picture)

This leads to the boundary conditions

which defines k: where has to be an integer (). This means that we actually have standing waves in the cube

Now we want to get the total energy. For this, we try to find the amount of waves per frequency interval of and . Later we will integrate over it. If we do this in spherical coordinates, the volume we integrate over equals of a total sphere in this cube (watch the picture – The visible faction of the sphere is of a total sphere).

First of all we need the wave density according to a specific frequency . We take the definition of k and n from the boundary conditions and transform this equation

So we get .

We also know from classical physics that the energy of a standing wave is proportial to where is the Boltzmann constant and T the temperature. All over we get the energy per volume .

This finally leads to the requested (infinitesimal) energy density . This shows us that it is proportial to .

If we integrate the found energy density over all frequencies in order to get the total energy we get

Interpolation of Planck

By experimenting with black bodies he found an empirical formula to describe the energy density:

or rearrangeg:

where is the planck constant divided by :

This is known as Planck’s Law. It interpolates between the already known Rayleigh-Jeans law for low energy scales and Wien’s approximation for higher energy scales.

To find the relation between the quanta and the energy we need to derive Planck’s Law directly instead of finding it empirical out of experiments.

For this derivation the idea of the quanta is used.

First of all we define the probability for a single energy. This is . Here we used the the Boltzmann factor to derive this formula. Our formula is similar to the Boltzman distribution of thermodynamical systems.

This leads to an expression for the average energy: . The formula is easy to understand: We sum over all energies multiplied by their probability.

Because we said that the energie has to be emitted or absorbed as packages we define (the idea of the quanta!) where is a multiplier and that mentioned tiniest energy package (quantum).

Then after inserting every formulas this leads to

Now we use a mathematical trick: The value of the convergent, infinite geometric series is if z is smaller than 1 and greater than 0 (if z isn’t smaller than 1, this series is divergent and rises against infinity)

In the next step we first of all substitute to make the formulas easier to write.

We go on looking at the denominator of our total energy formula: . This is a valid operation because is smaller than 1 for every x. We also used the mathematical trick with the geometric series.

Now we want to do the same with the numerator. This part is a bit more tricky. We can write . We just excluded the constant out of the summands and then we used the definition of the derivation .

Rewriting this part again like we did with the denominator leads to

Again we used the trick with the geometrick series.

The total fraction now looks the following way:

This is exactly Planck’s law if you choose . This is our tiniest energy package we searched for.