What Is the Effective Temperature in a Population-Inverted Two-Level System?

[bon]

Homework Statement

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. Now it is clear that this system 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 that i have electrons that populate the two states in the normal manner at room temperature. I then somehow swap the populations (i/e/ all the ones that were in thw lower temperature 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 state?

Homework Equations

The Attempt at a Solution

not sure where to begin! any help would be great. thanks

 

[Maxim Zh]

Use Gibbs distribution.

 

[gamma5772]

Begin by solving a 2 state system at some temperature T. You may do this in the microcanonical ensemble or more easily, in the canonical ensemble.

You should find that, if the states are separated by an energy E, that the population in the higher energy state is

1/(1 + exp(-E/kT)).

You should be able to handle it from there.