Consider a particle moving freely inside a cubic box of side L, volume V = L3. We take the potential to be zero inside the box, and represent the walls by infinite potential barriers. The Schrodinger equation for the particle,

is separable in (x, y, z) coordinates, and the solutions must vanish at the walls which we can take to be the planes x = 0 and x = L, y = 0 and y = L, z = 0 and z = L. These solutions are easily seen to be standing waves of the form

ѱ (x, y, z) = (constant) sin(kxx) sin(kyy) sin(kzz), (B.2)

provided that we choose k = (kx, ky, kz) from the values

to satisfy the boundary conditions. Negative integer values of nx, ny, nz do not give new states, since they merely change the sign of the wave-function, and such a phase factor has no physical significance.

Thus the allowed values of k form a cubic lattice of points in the (+, +, +) quadrant of ‘k-space’. Each eigenstate (B.2) corresponds to one point of the lattice, and counting states is equivalent to counting lattice points. The spacing between these lattice points is (ℼ/L), so that the number of points per unit ‘volume’ in k-space is (L/ℼ)3. The number of lattice points with k(= |k|) less than some fixed value k0 is the number enclosed within the quadrant of the sphere centred at the origin and of radius k0.