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A Classical gas with general dispersion relation

  1. Apr 16, 2017 #1
    i'm trying to understand the solution to this problem:

    http://physweb.bgu.ac.il/COURSES/StatMechCohen/ExercisesPool/EXERCISES/ex_2065_sol_Y13.pdf
    (link to the problem and the solution of it)

    upload_2017-4-16_13-6-46.png

    All my questions come from the partition function:

    upload_2017-4-16_13-7-3.png

    1) From where the term (2*pi)^d comes from?, I think is like a normalization factor, but i'm not sure.


    2) The Volume (V) should be V^d, because is the volume of a particle of d dimension, but in the solution is just "V" , I don't understand why.


    3) The solid angle is used to simplify the integral and it comes from the volume of a sphere of d-dimensionm. I don't understand how to use that volume of the sphere to this specific problem.

    4) Where is this problem used, or is it just a theoretical problem?

    5) The last question is a conceptual one, how the phase space looks in a d-dimension, I don't understand this concept.

    Any help in this questions will be appreciated.
    Thanks in advance.
     

    Attached Files:

    Last edited: Apr 16, 2017
  2. jcsd
  3. Apr 20, 2017 #2

    Charles Link

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    Homework Helper

    The way I learned it, (the Statistical Physics book by F. Reif), ## e^{ik_x x}=e^{i k_x(x+L_x)} ## for periodic boundary conditions. This means in counting states in k-space in 3 dimensions, you get the number of states ## \Delta N = \Delta^3 n=V \Delta^3 k/(2 \pi)^3 ##. (## k_x L_x=n_x 2 \pi ##, ## k_y L_y=n_y 2 \pi ##, etc. from the periodicity requirement). Since ## p=\hbar k ##, this will also put ah ## \hbar^3 ## in the denominator of the ## Z ## function which counts the states and multiplies by the Boltzmann factor ## e^{-E/(kT)} ##.
     
    Last edited: Apr 20, 2017
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