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The Derivation of the Planck Formula for Thermal Radiation

Albert Einstein developed a simple but effective analysis of induced emission and absorption of radiation along with spontaneous emission that can be
used to derive the Planck formular for thermal radiation.

Consider two energy levels for the molecules in a material. The lower of the two is denoted as E_{1} and the higher as E_{2}.
The probability of a transition from level 1 up to level 2 through induced absorption is assumed to be proportional to the energy density per unit frequency interval, (du/dν). Likewise
the probability of an induced transition from level 2 down to level 1 is assumed also to be proportional to (du/dν). These two probabilities are taken to be
B_{12}(du/dν) and B_{21}(du/dν), respectively, where B_{12} and B_{21} are constants. The probability of a
spontaneous emission is assumed to be a constant A_{21}.

Let N_{1} and N_{2} be the number of molecules in energy states 1 and 2, respectively. For equilibrium the number of transitions from 1 to 2
has to be equal to the number from 2 to 1; i.e.,