Energies of a Trapped Electron — Question 1

Question 1

The first excitation probes an energy gap rather than the ground energy

An electron occupies a well of width $L=1.00\,\mathrm{nm}$. 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. Use $hc=1240\,\mathrm{eV\,nm}$ and neglect recoil.

1. Find the ground energy and the first excitation energy.

2. Find the photon wavelength resonant with the first excitation.

Question 1 - Solution

The first excitation probes an energy gap rather than the ground energy

(a) Method and calculation

$$\boxed{E_1=C/L^2=.376\,\mathrm{eV}},\quad E_2=4E_1.$$$$\boxed{\Delta E_{21}=3E_1=1.128\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$E_\gamma=hc/\lambda=\Delta E_{21}\Rightarrow\lambda=hc/(3E_1).$$$$\boxed{\lambda=1099.29\,\mathrm{nm}}.$$

Physical check

The photon supplies the difference between two levels. Using the ground-state energy itself as the absorption energy would identify the wrong wavelength.