Energies of a Trapped Electron — Question 10

Question 10

Resonant absorption depends on the initial mode as well as photon energy

A monochromatic photon has energy $5E_1$. An infinite well has levels $E_n=n^2E_1$. Consider single-photon absorption with negligible linewidth and recoil. Assume any transition satisfying the stated energy condition is otherwise allowed.

1. Determine whether absorption from the ground mode can reach another bound eigenstate.

2. Repeat for an electron initially in mode 2 and identify the final mode.

Question 10 - Solution

Resonant absorption depends on the initial mode as well as photon energy

(a) Method and calculation

$$n_i=1:\quad n_f^2E_1=E_1+5E_1\Rightarrow n_f^2=6.$$$$\sqrt6\notin\mathbb N\Rightarrow\boxed{\text{no resonant final box eigenstate}}.$$

(b) Interpretation and further calculation

$$n_i=2:\quad n_f^2E_1=4E_1+5E_1=9E_1.$$$$\boxed{n_f=3,\quad E_3-E_2=5E_1}.$$

Physical check

Energy conservation is necessary for resonance. Actual transition strengths also depend on coupling matrix elements; those are excluded from this energy-only model.