# Entropy of a generalised two-state quantum system

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1. Apr 17, 2017

### Gabriel Maia

Hi. This is the problem I'm trying to solve:

A system may be in two quantum states with energies '0' and 'e'. The states' degenerescences are g1 and g2, respectively. Find the entropy S as a function of the Energy E in the limit where the number of particles N is very large. Analyse this dependence and show that there is a region of negative temperature.

The energy of the system is given by

$$E=N(0\cdot P(0)+e P(e))$$

where,

$$P(0) = \frac{g1}{g1+g2} \,\,\,\,\,\,\,\,\,\, \mathrm{and} \,\,\,\,\,\,\,\,\,\, P(e) = \frac{g2}{g1+g2}$$

are the probabilities of the state with energy 0 and energy e being occupied. Now, I'm trying to find the number of possible microstates in order to calculate the entropy. Since I want to organise N particles in two groups of identical states, the number of microstates should be

$$\Sigma = \frac{N!}{\left(\frac{Ng1}{g1+g2}\right)!\left(\frac{Ng2}{g1+g2}\right)!}$$

Is this last expression correct? I'm not sure if it should be this or just the degenerescences.
Thank you very much.

2. Apr 22, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.