Spinless cubic-box wave-vector counting
= Spinless cubic-box wave-vector counting
Periodic boundaries on a cube give one spinless state per $(2\pi/L)^3$ in wave-vector space. Counting the full spherical shell gives $d\mathcal N/V=k^2dk/(2\pi^2)$. Hard-wall counting in the positive octant gives the same leading <volume> term.