Multiple Electrons in Rectangular Traps — Question 3

Question 3

Full spin polarization increases the confinement energy

Six noninteracting electrons occupy a one-dimensional well with energies $E_n=\epsilon n^2$. Compare the lowest orbital energy with unrestricted spins to the lowest energy when all six spins are forced parallel. Omit the energy needed to produce the polarization.

1. Find the two occupied-level patterns and energies.

2. Find the energy increase and the ratio of highest occupied energies.

Question 3 - Solution

Full spin polarization increases the confinement energy

(a) Method and calculation

$$\text{Unrestricted: }n=1,2,3\text{ twice}\Rightarrow\boxed{E_u=2\epsilon(1+4+9)=28\epsilon}.$$$$\text{Parallel: }n=1,\ldots,6\text{ once}\Rightarrow\boxed{E_p=\epsilon(1+4+9+16+25+36)=91\epsilon}.$$

(b) Interpretation and further calculation

$$\boxed{E_p-E_u=63\epsilon}.$$$$\boxed{E_{{\mathrm{highest}},p}/E_{{\mathrm{highest}},u}=36/9=4}.$$

Physical check

Parallel spin removes the second spin slot in each spatial orbital. The increase arises from exclusion and confinement even without an interelectron repulsive potential.