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!

Homework Help: Density of states in D dimensions

  1. Nov 23, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the density of states g(ε) for an ideal quantum gas of spinless particles in dimension d with dispersion relation  ε= α|p|s , where ε is the energy and p is the momentum of a particle. The gas is confined to a large box of side L (so V = Ld) with periodic boundary conditions. The density of states is defined as the number of single particle energy states with energy between ε and ε + dε. You can use the volume of a d-dimensional sphere of radius R,
    Ω0 = 2πd/2/dΓ(d/2)Rd.

    3. The attempt at a solution

    In 3D we solved this problem by solving the Schrodinger eq. where ε~p2
    but what happen when the dispersion relation is ε= α|p|s?

    My attempt was to define Γ(ε) as the number of states with energy ≤ ε
    in d- dimentions Γ(ε) = the volume of a d-dimensional sphere of radius n(ε) (n ia the quantum num)

    but how can i find the relation between n and ε?

    is it ok to say : p=(h/2π)k and k=πn/L and then just put it in the dispertion relation?

    Thank you!
  2. jcsd
  3. Nov 23, 2013 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    Yes, but k=πn/L is not the correct relation for periodic boundary conditions. (Darn factors of 2.)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted