Suppose that by some artificial means it is possible to put more electrons in the higher energy state than in the lower energy state of a two level system (this sort of situation occurs in a laser, for example). Now it is clear that this cannot be an equilibrium
situation, but nevertheless, for the time that the system is in this strange state we could, if we wished, still express the ratio of the populations in the upper and lower states by some parameter we can think of as an effective temperature.

(i) Show that for such a population inversion to exist, the effective temperature must be negative.

(ii) Imagine I have electrons that populate two states in the normal manner at room temperature. I then somehow swap the populations (i.e. all the ones that were in the lower state go into the upper state, and vice versa). What is the new effective temperature?

(iii) What is the effective temperature if I put all the electrons in the upper energy state?