Three-level Quantum System

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AlexTab
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Summary:: Find the ratio of the number of particles on the upper level to the total number in the system.

Consider an isolated system of ##N \gg 1## weakly interacting, distinct particles. Each particle can be in one of three states, with energies ##- \varepsilon_0##, ##0## and ##\varepsilon_0##. The energy of entire system is ##E##.

The temperature is defined as ##\displaystyle T = \frac{\partial S}{\partial E}##.

I need to find the ratio of the number of particles on the level with ##\varepsilon_0## to the total number ##N## in low temperature conditions ##T \ll \varepsilon_0##.


I start with finding the statistical weight of each state of the system as ##\displaystyle W = \frac{N!}{n_{-\varepsilon_0}! n_0! n_{\varepsilon_0}!}##, where ##n_{-\varepsilon_0}##, ##n_0## and ##n_{\varepsilon_0}## are the numbers of particles in the corresponding states. Then I can find the system entropy ##S = \ln W##.

Obviously, the number of particles at each next level is much lower than at the previous one.

It seems to me that the above should be enough to solve the problem, but I don't understand how to use these facts. Also there is a problem with the number of particles, because we don't know ##n_{-\varepsilon_0}## and ##n_{\varepsilon_0}##, only ##N## and ##\displaystyle n_{\varepsilon_0} -n_{-\varepsilon_0} = \frac{E}{\varepsilon_0}## are given.
 

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  • #2
Twigg
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It seems to me like this problem is having you re-invent the wheel, because what you're doing right now is deriving the canonical ensemble from stat mech.

Note that each tuple ##(n_+,n_0,n_-)## is one way of dividing the ##N## particles into the 3 energy levels, so let's call it a "distribution". You have a function for the entropy ##S = \ln W## that spits out the entropy associated with each distribution. If you leave an isolated system to do its thing for a while, what do you think the entropy of the system will do over time? If you wait until the system reaches steady state, do you think the entropy will be maximum or minimum?

I've already kind of spoiled the fact that you'll want to do an optimization procedure on the entropy to get the distribution ##(n_+,n_0,n_-)## at statistical equilibrium. Keep in mind that this is a constrained problem, since ##N## and ##E## are fixed. So you'll want to think about Lagrange multipliers. Also, you'll want to use Stirling's approximation a lot.

I hope that's helpful!

Edit: One more hint, the Langrange multiplier that you will get from the constraint on N can be ignored for this problem. In general, this Lagrange multiplier is the chemical potential from thermodynamics. It only matters when the particle number is allowed to vary. On the other hand, the Lagrange multiplier that comes from the constraint on E is essential. You'll see why when you solve for it.
 
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