Multiple Electrons in Rectangular Traps — Question 4

Question 4

A filled square-box shell includes degenerate spatial states

In a two-dimensional square of side L, $E_{n_xn_y}=\epsilon(n_x^2+n_y^2)$ with positive integer indices and two spins per spatial orbital. Six independent electrons occupy their zero-temperature ground state.

1. Identify occupied spatial orbitals and calculate the total energy.

2. Find the lowest energy to promote one electron into an unoccupied spatial orbital, ignoring transition selection rules.

Question 4 - Solution

A filled square-box shell includes degenerate spatial states

(a) Method and calculation

$$(1,1):2\epsilon;\quad(1,2),(2,1):5\epsilon\text{ each}.$$$$\boxed{E_g=2(2\epsilon)+4(5\epsilon)=24\epsilon}.$$

(b) Interpretation and further calculation

$$\text{Lowest empty orbital }(2,2):E=8\epsilon.$$$$\boxed{\Delta E_{\min}=8\epsilon-5\epsilon=3\epsilon}.$$

Physical check

Degenerate orbitals remain distinct states and each has its own two spin slots. The excitation cost is an energy difference, not the absolute energy of the empty orbital.