Multiple Electrons in Rectangular Traps — Question 2

Question 2

Adding one electron creates an even-odd pattern in the addition cost

Use noninteracting electrons with $E_n=\epsilon n^2$ and two spin states per spatial level. Define $A_N=E_g(N+1)-E_g(N)$ as the energy cost to add one electron into the same fixed infinite well.

1. Derive $A_{2K}$ and $A_{2K+1}$.

2. Find $A_5,A_6,A_7$ and distinguish this cost from ionization to free space.

Question 2 - Solution

Adding one electron creates an even-odd pattern in the addition cost

(a) Method and calculation

$$N=2K:\quad 1,\ldots,K\text{ filled}\Rightarrow\boxed{A_{2K}=\epsilon(K+1)^2}.$$$$N=2K+1:\quad\text{one vacancy in level }K+1\Rightarrow\boxed{A_{2K+1}=\epsilon(K+1)^2}.$$

(b) Interpretation and further calculation

$$\boxed{A_5=9\epsilon,\quad A_6=16\epsilon,\quad A_7=16\epsilon}.$$$$V_{\mathrm{outside}}=\infty\Rightarrow\boxed{\text{no finite escape threshold in this model}}.$$

Physical check

The addition cost compares different particle-number ground states in the same confinement. A physical ionization energy requires a finite exterior potential and an energy reference.