Multiple Electrons in Rectangular Traps — Question 7

Question 7

Dipole absorption is blocked when the destination spin orbital is filled

Six independent electrons fill $n=1,2,3$ in a one-dimensional well, two per spatial level. A weak electric-dipole perturbation preserves spin. Box matrix elements of x vanish for equal-parity indices and can be nonzero for opposite-parity indices. Energies are $E_n=\epsilon n^2$.

1. Find the lowest allowed one-electron absorption from the ground configuration.

2. Explain why the one-electron $1\to2$ line is absent and find the next lowest allowed absorption energy.

Question 7 - Solution

Dipole absorption is blocked when the destination spin orbital is filled

(a) Method and calculation

$$n=3\to4:\quad\Delta n=1,\quad n=4\text{ empty}.$$$$\boxed{\Delta E_1=(16-9)\epsilon=7\epsilon}.$$

(b) Interpretation and further calculation

$$1\to2:\quad\text{both destination spin orbitals occupied}\Rightarrow\boxed{\text{Pauli blocked}}.$$$$1\to4:\ 15\epsilon;\quad2\to5:\ 21\epsilon;\quad3\to6:\ 27\epsilon\Rightarrow\boxed{\Delta E_2=15\epsilon}.$$

Physical check

Both an allowed matrix element and an available final spin orbital are required. This independent-electron model excludes collective and multi-electron optical transitions.