How to Find Density of States for Quantum Gas in D Dimensions?

[Raz91]

Homework Statement

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 = L^d) 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 = 2π^d/2/dΓ(d/2)R^d.

The Attempt at a Solution

In 3D we solved this problem by solving the Schrödinger 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)
g(ε)=dΓ(ε)/dε

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!

 

[TSny]

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 dispersion relation?

Yes, but k=πn/L is not the correct relation for periodic boundary conditions. (Darn factors of 2.)