Energies of a Trapped Electron — Question 2

Question 2

Two adjacent spectral gaps identify both the level index and well width

Consecutive adjacent-level gaps in an infinite electron well are $E_{n+1}-E_n=1.00\,\mathrm{eV}$ and $E_{n+2}-E_{n+1}=1.40\,\mathrm{eV}$. 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. Infer n from the ratio of the gaps.

2. Find the ground energy and well width.

Question 2 - Solution

Two adjacent spectral gaps identify both the level index and well width

(a) Method and calculation

$$\frac{1.40}{1.00}=\frac{2n+3}{2n+1}\Rightarrow 2.8n+1.4=2n+3.$$$$\boxed{n=2}.$$

(b) Interpretation and further calculation

$$\boxed{E_1=1.00/(2n+1)=.200\,\mathrm{eV}}.$$$$\boxed{L=\sqrt{C/E_1}=\sqrt{1.88}\,\mathrm{nm}=1.37113\,\mathrm{nm}}.$$

Physical check

The ratio isolates the integer index independently of width. A ratio inconsistent with an integer would challenge the assumed box model or line assignment.