Two- and Three-Dimensional Electron Traps — Question 2

Question 2

Square-well degeneracy depends on sums of squares rather than a single index

In a rectangular infinite well, $E_{n_xn_y}=C(n_x^2/L_x^2+n_y^2/L_y^2)$, where $C=\pi^2\hbar^2/(2m)$ and each index is a positive integer. Ignore spin. Let $L_x=L_y=L$ and $A=C/L^2$.

1. List all states at energies $5A$ and $8A$.

2. List all states at $10A$ and give the spatial degeneracy of each energy.

Question 2 - Solution

Square-well degeneracy depends on sums of squares rather than a single index

(a) Method and calculation

$$n_x^2+n_y^2=5\Rightarrow\boxed{(1,2),(2,1)}.$$$$n_x^2+n_y^2=8\Rightarrow\boxed{(2,2)}.$$

(b) Interpretation and further calculation

$$n_x^2+n_y^2=10\Rightarrow\boxed{(1,3),(3,1)}.$$$$\boxed{g(5A)=2,\quad g(8A)=1,\quad g(10A)=2}.$$

Physical check

Interchanging unequal indices produces distinct states at equal energy in a square. Equal indices produce only one state, and zero indices are excluded.