Translational Degeneracy

The energy levels of a particle in 3-dimensional box are as follows

E = h²/8ma²(n1² + n2² + n3²) = h²/8ma²· K² = E1 · K²

All combinations n1, n2 and n3 that give finally the same K value have the same energy and therefore it's said these energy levels are degenerated. This degeneracy degree which is assigned by quantity g corresponds to amount of independent wavefunctions.
The amount of energy levels within thin energy region dE can be calculated now for a particle situated in a big potential well. The states amount N(E) beginning from zero energy and till energy E corresponds to sphere volume that has radius K (K² = n1²+n2²+n3²) divided by 8 since only positive values of n1, n2 and n3 are resolved:

N(E) = 1/8(4/3πK3) = π/6 V (8mE/h²)3/2

The second equality sign is obtained from E = h²/8ma²·K² and K= a·(8mE/h²)1/2 and V = a³. Differentiating the last formula by dE, one can obtain the number of states in the range [E, E+dE]:

d N(E) = [4πV(2m³)½/h³] E½ dE

The energy levels density g(E) = dN/dE, i.e. the number of states per energy unit range is then as follows:

g(E) = 4πV(2m³)½/h³. E½

Here we've just obtained the statistical sum:

Q = oò¥ e−E/kT g(E) dE
Q = 4πV/h³ (2m³)½ oò¥ E½ e−E/kT dE
ß

Q = V/h³ (2πmkT)3/2

Moreover we've obtained very important quantity namely the number of state translations. One can get all thermodynamical quantities (relatively translation) from the calculations considered.