What Is the Effective Temperature in a Population Inversion Scenario?

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In a population inversion scenario, the effective temperature is defined as negative when there are more electrons in a higher energy state than in a lower one, indicating a non-equilibrium condition. To explore this, one must analyze the population ratios using the Boltzmann factor. If electrons are swapped between states at room temperature, the new effective temperature can be calculated based on the altered population distribution. Additionally, if all electrons are placed in the upper state, the effective temperature would also need to be determined based on this complete inversion. Understanding these concepts is crucial for grasping the implications of population inversion in quantum systems.
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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 really even sure where to begin! Any help would be great..thanks! :)
 
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any ideas? I am guessing it has something to do with the Boltzmann factor..?
 
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