This post is the solution of the following problem: pick an atom with two energy levels. They have a transition wavelength of 580nm. At room temperature are in the lower state.
1) How many atoms are in the upper state, if we have thermal equilibrium?
2) Suppose instead we had atoms pumped into the upper state, with atoms in the lower state. This is a non-equilibrium state. How much energy (in Joules) could be released in a single pulse of light if we restored the equilibrium?
Data: Boltzmann factor is and
Solution to 1). If is the higher energy state and is the lower energy state, then the energy difference between these two atomic states can be calculated using the transition wavelength (580nm), i.e.,
Now, according to the Boltzmann distribution, the population in any state is given by
Therefore, the ration of and in the thermal equilibrium should be
Using the giving value of , we get
and that is a very tiny number of atoms. This is showing to us that the energy gap between the given atomic states at room temperature is so large that almost all the electrons “choose” to stay in the lower state and hardly we will find electrons in the upper state.
Solution to 2). When we pump atoms into the upper state, we create a non-equilibrium atomic state. Energy will be released up until the equilibrium is restored (in fact, this is the working principle of the laser, in a simplified fashion!). We obtain now
and we are interested in the number of atoms which restores the thermal equilibrium. Since the ratio should be
remains the same (the gap width should not change), we find
The number of atoms that contribute to restore the thermal equilibrium is
The energy released in a single monochromatic pulse would be