1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    [tex]
    n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } (e^{-n\varepsilon \beta})
    [/tex]
    and use the fact that the partial derivative commutes with the summation to get
    [tex]
    \sum_{n=0}^{N} n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } \sum_{n=0}^{N} e^{-n\varepsilon \beta}
    [/tex]
    The summation is now simply just a geometric series.
    (and don't forget to normalize your expectation value)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Average energy for n-state system
Loading...