Particle in 3D Box: Relation between Ω(E) and E

[charbon]

Homework Statement

For a system consiting of a single particle of mass m in a box of volume L^3 (Lx = Ly = Lz = L) develop a relation between the number of accessible states, Ω(E) and E

Homework Equations

E = ((π^2ћ^2)/(2mL^2))(nx^2 +ny2 +nz2)

The Attempt at a Solution

nx^2 + ny^2 + nz^2 = (2mEL^2)/(π^2ћ^2)

this is the equation of a sphere. The next step would be to find the number of states with energy inferior to E (ψ(E)) but I'm a bit clueless about how to do that with the equation. Could someone clarify that for me? Thanks in advance

 

[fzero]

If you can assume that the radius of the sphere is much larger than 1, you can use a volume integral to approximate the sum over states. You must be careful that your counting respects that the n_i>0.