Two- and Three-Dimensional Electron Traps — Question 3

Question 3

Changing aspect ratio can tune two different modes into degeneracy

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.

1. Find $L_y/L_x$ such that states (1,3) and (2,1) have equal energy.

2. At fixed $L_x$, determine which of these two states has lower energy on each side of that aspect ratio.

Question 3 - Solution

Changing aspect ratio can tune two different modes into degeneracy

(a) Method and calculation

$$\frac1{L_x^2}+\frac9{L_y^2}=\frac4{L_x^2}+\frac1{L_y^2}\Rightarrow\frac8{L_y^2}=\frac3{L_x^2}.$$$$\boxed{L_y/L_x=\sqrt{8/3}=1.63299}.$$

(b) Interpretation and further calculation

$$E_{13}-E_{21}=\frac C{L_x^2}\left[\frac8{(L_y/L_x)^2}-3\right].$$$$\boxed{L_y/L_x>\sqrt{8/3}\Rightarrow E_{13}<E_{21}},\quad\boxed{L_y/L_x<\sqrt{8/3}\Rightarrow E_{13}>E_{21}}.$$

Physical check

This crossing follows from separability of the ideal rectangular well. A perturbation coupling the modes can turn a crossing into an avoided crossing.