Two- and Three-Dimensional Electron Traps — Question 1

Question 1

The longer side controls the lowest excitation of a rectangular well

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 $L_x=L$, $L_y=2L$ and $A=C/L^2$.

1. Find the ground energy and the energies of states (2,1) and (1,2).

2. Identify the first excited state and the smallest energy gap.

Question 1 - Solution

The longer side controls the lowest excitation of a rectangular well

(a) Method and calculation

$$\boxed{E_{11}=A(1+1/4)=5A/4}.$$$$\boxed{E_{21}=17A/4,\qquad E_{12}=2A}.$$

(b) Interpretation and further calculation

$$E_{12}<E_{21}\Rightarrow\boxed{\text{first excited state }(1,2)}.$$$$\boxed{\Delta E_{\min}=E_{12}-E_{11}=3A/4}.$$

Physical check

Exciting the mode along the longer side costs less energy. The ground state still has a positive contribution from both directions.