- #1

Raz91

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## 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 Schrodinger eq. where ε~p

^{2}

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!