Multiple Electrons in Rectangular Traps — Question 8

Question 8

An elongated rectangle changes the order in which orbitals fill

A rectangular well has $L_x=2L$, $L_y=L$ and energies $E=\epsilon(n_x^2/4+n_y^2)$, with $\epsilon=\pi^2\hbar^2/(2mL^2)$. Six noninteracting electrons occupy the ground state.

1. Find the three lowest spatial orbitals and their energies.

2. Calculate the total ground energy and the first excitation gap.

Question 8 - Solution

An elongated rectangle changes the order in which orbitals fill

(a) Method and calculation

$$\boxed{E_{11}=5\epsilon/4,\quad E_{21}=2\epsilon,\quad E_{31}=13\epsilon/4}.$$$$E_{12}=17\epsilon/4<E_{41}=5\epsilon\Rightarrow\text{next empty orbital }(1,2).$$

(b) Interpretation and further calculation

$$\boxed{E_g=2\epsilon(5/4+2+13/4)=13\epsilon}.$$$$\boxed{\Delta E_{\min}=E_{12}-E_{31}=\epsilon}.$$

Physical check

Filling requires comparing all candidate directions. Increasing the long-axis quantum number is initially cheaper, but eventually exciting the short direction becomes favorable.