Concept of Degeneracy

An interesting result is obtained about the energies of the particle in a three dimensional box when the sides of the box are equal i.e., it is a cubical box. For such a case, putting a=b=c, eqn. becomes
                                        
The different combinations of the values of quantum number n_x, n_y and n_z may give the same value of the energy E. For example, the following six combinations of the values 1,2 and 3 give the same value of E

n_x                   1           1           2           2             3            3
n_y                   2          3           1           3              1            2
n_z                   3          2           3            1             2            1

Such an energy level having different states of the system (different wave function*) but having the same energy is said to be degenerate and the number of independent wave functions associated with the given energy level is called its degeneracy.

The degeneracy’s of a few energy levels with different quantum numbers are given below:

n_x , n_y , n_z           111            211           221        311         222           321         332
Degeneracy         1                  3               3            3               1               6              3

As an example, the three states corresponding to the quantum numbers 211 will have the quantum numbers, 211, 121 and 112. All these states will have energy =   (obtained by putting the values of n_x , n_y and n_z in eqn.

It is important to mention here that degeneracy’s arise in quantum mechanics when there is some element of symmetry. If the symmetry is broken e.g. by taking a box whose sides have different lengths, the degeneracy vanishes.

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