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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 nx, ny and nz 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

nx                   1           1           2           2             3            3
n                  2          3           1           3              1            2
nz                   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:

nx , ny , nz           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 nx , ny and nz 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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