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Homework Help: Average energy for n-state system

  1. Jan 26, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the average energy ##\langle E \rangle## for
    (a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε.
    (b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit).

    2. Relevant equations
    Definition of temperature: ##β = \frac 1 {K_BT} = \frac {d lnΩ(E)} {dE}##
    Boltzmann distribution: ##P(ε) ∝ e^{-εβ}##

    3. The attempt at a solution
    Since the energy here takes on discrete values, the average is found by taking the weighted sum of the probabilities, $$\langle E \rangle = \sum_{n=0}^n nε~e^{-nεβ}$$
    and in the case of part (b), the sum goes to infinity. My problem is I don't know how to evaluate these sums. Any help would be appreciated, thanks!
  2. jcsd
  3. Jan 27, 2017 #2
    One commonly used trick to evaluate such sums is the following observation that
    n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } (e^{-n\varepsilon \beta})
    and use the fact that the partial derivative commutes with the summation to get
    \sum_{n=0}^{N} n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } \sum_{n=0}^{N} e^{-n\varepsilon \beta}
    The summation is now simply just a geometric series.
    (and don't forget to normalize your expectation value)
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