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

  1. Dec 6, 2011 #1
    Consider a system S in contact with a reservoir R at a temperature [itex]\tau[/itex]. 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 [itex]\epsilon_i[/itex]. Your probability should be normalized. Define any quantities you introduce.

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




    2. Relevant equations

    Not sure but I think it ought to involve

    Partition function: [itex]Z = \sum_i e^{-\epsilon_i/\tau}[/itex]

    [itex]P(\epsilon_i)=\frac{e^{-\epsilon_i/\tau}}{Z}[/itex]

    3. 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 [itex]\epsilon[/itex], rather than the probability that the system has energy [itex]\epsilon_i[/itex]. 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 [itex]U=\tau^2 \frac{\partial \log Z}{\partial \tau}[/itex] if that helps any. But those parts I think I can get once I know what we are doing.
     
  2. jcsd
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