Multiple Electrons in Rectangular Traps — Question 1

Question 1

An even number of box electrons fills a sum of squares

Place $N=2K$ noninteracting spin-half electrons in a one-dimensional infinite well of length L. The spatial energies are $E_n=\epsilon n^2$, where $\epsilon=\pi^2\hbar^2/(2mL^2)$. At zero temperature each spatial level holds two electrons. Use $\sum_{n=1}^Kn^2=K(K+1)(2K+1)/6$.

1. Derive the ground-state total energy.

2. For eight electrons find the total energy and highest occupied one-electron energy.

Question 1 - Solution

An even number of box electrons fills a sum of squares

(a) Method and calculation

$$E_g=2\epsilon\sum_{n=1}^Kn^2.$$$$\boxed{E_g=\epsilon K(K+1)(2K+1)/3}.$$

(b) Interpretation and further calculation

$$N=8\Rightarrow K=4:\quad\boxed{E_g=60\epsilon}.$$$$\boxed{E_{\mathrm{highest}}=E_4=16\epsilon}.$$

Physical check

The total energy is a sum over occupied spin orbitals, whereas the highest occupied energy belongs to just one electron. The model omits electron-electron interactions.