Calculating the Number of States: Ω = V^N * E^0.5N

[pallab]

Homework Statement

no of states for N identical free particles with energy between E and E+δE is proportional to -

Relevant Equations

for microcanonical ensemble

the answer is NE^0.5 but my answer is E^0.5N
the # of state is Ω=( one particle phase-space volume)^N
one particle phase-space volume=integration of dq*integration of dp
from space part dq I get V and dp is converted into Energy E variable via E=p^2/2m

 

[mitochan]

considering 3D and the volume of sphere in 3N dimensional space
$$\sum_{i=1}^n (p_{ix}^2+p_{iy}^2+p_{iz}^2)=2mE$$
I got volume of the shell is proportional to
$$E^{\frac{3N}{2}-1}dE$$