Mode note: moved to HW help

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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?

 

 

I'm quite puzzled by this. Could someone please explain this or tell me the formulae I need to use? Thanks a lot.

 

 

I have also attached the problem set I'm trying to solve. I couldn't get questions 2.4 and 2.2 as well so I'd welcome answers or explanations of those as well. Thanks.

sm-questions.pdf

The only formula you need is the relative probability for a particle to be in a state with energy E as a function of the temperature. It is [math] \rho(E) = n_E \exp (-\beta E) [/math], with [math] \beta = \frac{1}{k_B T} [/math] and n_E being the number of states with an energy E (you probably want n_E = 1 in above).

 

EDIT: Please note that this forum has a "homework help" section where your posts are probably more appropriate. Also note the rules applying for this forum (should be written in a pinned thread inside that subforum).