Probability a system S will be in a quantum state i with energy ei. (Theory)

[anotherghost]

Consider a system S in contact with a reservoir R at a temperature $\tau$. The number of particles N and the volume V are fixed.

A: Give the probability that S will be in a quantum state i with energy $\epsilon_i$. Your probability should be normalized. Define any quantities you introduce.

B: If there are n different quantum states with the same energy $\epsilon$, what is the normalized probability S will have an energy $\epsilon$?

Homework Equations

Not sure but I think it ought to involve

Partition function: $Z = \sum_i e^{-\epsilon_i/\tau}$

$P(\epsilon_i)=\frac{e^{-\epsilon_i/\tau}}{Z}$

The Attempt at a Solution

I honestly don't even know where to start. Our lecturer says we went over this in class but we really didn't. A shove in the right direction would be really appreciated.

What is confusing me about it is the wording. We want the probability that the system S is in a particular quantum state i, which happens to have energy $\epsilon$, rather than the probability that the system has energy $\epsilon_i$. I don't know how to do this without more information.

Ultimately, in later parts of the question, we have to disattach and reattach the reservoirs, then derive the expression $U=\tau^2 \frac{\partial \log Z}{\partial \tau}$ if that helps any. But those parts I think I can get once I know what we are doing.

 

[jbsteele]

A: The probability that S will be in the quantum state i with energy \epsilon_i is given by the expression: P(\epsilon_i) = \frac{e^{-\epsilon_i/\tau}}{\sum_i e^{-\epsilon_i/\tau}}. This expression is normalized, meaning that the probability of being in any quantum state is guaranteed to be between 0 and 1, and the sum of all the probabilities of being in any quantum state is equal to 1.B: The probability that S will have an energy \epsilon is given by the expression: P(\epsilon) = \frac{1}{n}\sum_{i \in \epsilon} \frac{e^{-\epsilon_i/\tau}}{\sum_i e^{-\epsilon_i/\tau}}, where n is the number of states with energy \epsilon. This expression is also normalized, guaranteeing that the probability of having an energy \epsilon is between 0 and 1 and the sum of all the probabilities of having any energy is equal to 1.