Two- and Three-Dimensional Electron Traps — Question 6

Question 6

Three-dimensional degeneracies count distinct permutations of positive indices

In a three-dimensional rectangular infinite well, $E=C(n_x^2/L_x^2+n_y^2/L_y^2+n_z^2/L_z^2)$, with positive integer indices and $C=\pi^2\hbar^2/(2m)$. Ignore spin. For a cube of side L define $A=C/L^2$.

1. Find all states with energy $14A$ and their degeneracy.

2. Repeat for energy $12A$ and explain the different counts.

Question 6 - Solution

Three-dimensional degeneracies count distinct permutations of positive indices

(a) Method and calculation

$$n_x^2+n_y^2+n_z^2=14=1+4+9\Rightarrow\{n_x,n_y,n_z\}=\{1,2,3\}.$$$$\boxed{g(14A)=3!=6}\quad\text{distinct permutations}.$$

(b) Interpretation and further calculation

$$n_x^2+n_y^2+n_z^2=12=4+4+4\Rightarrow\boxed{(n_x,n_y,n_z)=(2,2,2)}.$$$$\boxed{g(12A)=1}.$$

Physical check

The allowed indices are positive integers. Spin degeneracy is excluded, and the permutation count must be reduced when indices repeat.