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!

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

    User Avatar
    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
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: Classical gas with general dispersion relation
  1. Dispersion Relations (Replies: 1)

  2. Classical Ideal Gas (Replies: 5)

Loading...