What Is Degeneracy in Quantum Mechanics?

[phyzmatix]

First of all, I just want to check if my understanding of degeneracy is correct:

The degeneracy of an excited state is the number of combinations of quantum numbers that will result in the same energy level.

?

Secondly, if this is right and if we have an equation

$$E_{xy}=\frac{\hbar^2 \pi^2}{2ma^2}(n_x^2+n_y^2)$$

from which we wish to obtain the four lowest possible energy levels and their corresponding degeneracies, then is it correct to do that as follows

$$E_{11}=\frac{\hbar^2 \pi^2}{2ma^2}(1^2+1^2)=2\frac{\hbar^2 \pi^2}{2ma^2}$$

$$E_{12}=\frac{\hbar^2 \pi^2}{2ma^2}(1^2+2^2)=5\frac{\hbar^2 \pi^2}{2ma^2}$$

$$E_{21}=\frac{\hbar^2 \pi^2}{2ma^2}(2^2+1^2)=5\frac{\hbar^2 \pi^2}{2ma^2}$$

$$E_{22}=\frac{\hbar^2 \pi^2}{2ma^2}(2^2+2^2)=8\frac{\hbar^2 \pi^2}{2ma^2}$$

With corresponding degeneracies

$$E_{11} = 1$$
$$E_{12}=E_{21}=2$$
$$E_{22}=1$$

I'm not sure if there's an equation that could be useful to calculate the degeneracy perhaps?

Thanks in advance!
phyz

 

[graphene]

The degeneracy of an energy level is the number of different states having that energy.
(you better say state than quantum numbers).

How much of quantum mechanics have you done?
Do you know how the calculate the eigenvalues and eigenstates, given the matrix of an operator?
That is how you find out degeneracies.