Two- and Three-Dimensional Electron Traps — Question 9

Question 9

At fixed area the square minimizes the rectangular ground-state energy

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. The area $\mathcal A=L_xL_y$ is fixed while the aspect ratio may vary.

1. Find the geometry minimizing the ground energy and its minimum.

2. Compare the ground energy for $L_y=2L_x$ with this minimum.

Question 9 - Solution

At fixed area the square minimizes the rectangular ground-state energy

(a) Method and calculation

$$E_{11}=C\left(\frac1{L_x^2}+\frac{L_x^2}{\mathcal A^2}\right)\ge\frac{2C}{\mathcal A}.$$$$\boxed{L_x=L_y=\sqrt{\mathcal A},\qquad E_{\min}=2C/\mathcal A}.$$

(b) Interpretation and further calculation

$$L_y=2L_x:\quad L_x^2=\mathcal A/2\Rightarrow E_{11}=C(2+1/2)/\mathcal A.$$$$\boxed{E_{11}/E_{\min}=5/4}.$$

Physical check

Concentrating confinement into one short direction costs more energy than the long direction saves. This comparison holds at fixed area, not fixed perimeter.