Energies of a Trapped Electron — Question 3

Question 3

A radiative cascade conserves total photon energy but not photon wavelength sums

An electron starts at $n=4$ in a $1.00\,\mathrm{nm}$ infinite well and ends at $n=1$. Compare a direct transition with the cascade $4\to3\to2\to1$, assuming the transitions occur and recoil is negligible. Model the electron by a one-dimensional infinite well: $E_n=Cn^2/L^2$, $n=1,2,\ldots$, with $C=.376\,\mathrm{eV\,nm^2}$ when L is in nanometers. Set the interior potential to zero.

1. Find the three cascade photon energies and their sum.

2. Compare the mean cascade photon energy with the single direct photon energy.

Question 3 - Solution

A radiative cascade conserves total photon energy but not photon wavelength sums

(a) Method and calculation

$$(E_{43},E_{32},E_{21})=E_1(16-9,9-4,4-1)=E_1(7,5,3).$$$$\boxed{(E_{43},E_{32},E_{21})=(2.632,1.880,1.128)\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\mathrm{total}}=15E_1=5.640\,\mathrm{eV}=E_{41}}.$$$$\boxed{\overline E_{\mathrm{cascade}}=E_{41}/3=1.880\,\mathrm{eV}}.$$

Physical check

Energy differences telescope along the cascade. Wavelengths are inversely proportional to photon energies, so wavelength sums do not obey this conservation rule.