Translational Degeneracy

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

E  =  /8ma²(n1² + n2² + n3²)  =  /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/)3/2

The second equality sign is obtained from E = /8ma²·K² and K= a·(8mE/)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/] 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/. E½

Here we've just obtained the statistical sum:

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