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Homework Help: Fermions that can access 10 distinct energy states; Statistical Physics

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a system made of 4 quantum fermions that can access 10 distinct states respectively with energies:

    En=n/10 eV with n=1,2,3,4,5,6,7,8,9,10

    1) Write the expression for the entropy when the particles can access all states with equal probability

    2) Compute the Entropy of the isolated system at energy U =1 eV

    3) Compute the entropy of the isolated system at energy 1.1 eV

    2. Relevant equations

    Ω=G!/m!(G-m)!
    s=kBln(Ω)


    3. The attempt at a solution

    the first question i think is answered basically by the first equation i gave for the statistical weight because that is for indistinguishable particles with multiple occupancy not allowed. Im a little bit stuck on the 2nd and 3rd questions. The probability of finding a particle in the lowest state must be more probable than finding a particle in the highest state but the equation for the statistical weight wont take that into account. if i can be pointed in the right direction that would be awesome :smile:
     
  2. jcsd
  3. Apr 16, 2012 #2
    The physical way of looking at entropy is that it's the logarithm of the number of states corresponding to a given energy. For example, when the total energy is fixed to 1eV, there's only one possible arrangement to get this: 1/10eV + 2/10eV + 3/10eV + 4/10eV = 1eV. Now it becomes just a combinatorics problem, and you only need to figure out how many such configurations there are.
     
  4. Apr 17, 2012 #3
    ok so because there is only 1 possible arrangement for u=1ev the statistical weight can be calculated used the equation above so

    Ω= 4!/4!(1)!
    Ω=1

    so s=kln(1)

    then for the u=1.1 ev so the only possible state will be when 3/10 + 5/10 + 2/10 + 1/10 = 1.1 ev

    so again Ω=1

    s=kln (1)
     
  5. Apr 18, 2012 #4
    Oops, I forgot about spin. If your fermions have spin 1/2, you can have two of them occupying a state with same n. Maybe you should at least mention this, if not calculate it completely.
     
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