An Electron in a Finite Well — Question 10

Question 10

Measured oscillation and decay scales reconstruct a finite-well ground state

A symmetric finite electron well has an even nodeless bound state with measured interior wave number $k=4.00\,\mathrm{nm}^{-1}$ and exterior decay constant $\kappa=3.00\,\mathrm{nm}^{-1}$. Use $k\tan(ka)=\kappa$, $E=\hbar^2k^2/(2m_e)$ and $V_0-E=\hbar^2\kappa^2/(2m_e)$, with $\hbar^2/(2m_e)=.0381\,\mathrm{eV\,nm^2}$.

1. Find E and $V_0$.

2. Infer the full width, choosing the correct branch for the ground state.

Question 10 - Solution

Measured oscillation and decay scales reconstruct a finite-well ground state

(a) Method and calculation

$$\boxed{E=.0381(4.00)^2=.6096\,\mathrm{eV}}.$$$$\boxed{V_0=.0381(4^2+3^2)=.9525\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$0<ka<\pi/2\Rightarrow ka=\arctan(\kappa/k)=\arctan(.750).$$$$\boxed{L=2a=\frac{2\arctan(.750)}{4.00}\,\mathrm{nm}=.321751\,\mathrm{nm}}.$$

Physical check

Adding integer multiples of pi to the inverse tangent would describe other branches. Nodeless ground-state information removes that ambiguity.