Question

An electron in a finite potential well has a $1.0\mathrm{nm}$ penetration distance into the classically forbidden region. How far below ${\mathrm{U}}_0$ is the electron’s energy?

Short answer

The energy is below the$0.38\mathrm{eV}.$

Step-by-step solution

Given information 

We have given,

penetration depth =$1\mathrm{nm}$

We have to find the how below its energy.

Simplify

We know the penetration depth is,

$$\begin{gathered} \mathrm{\eta }=\frac{\mathrm{h}}{2\mathrm{\pi }\sqrt{2\mathrm{m}({\mathrm{U}}_{\mathrm{o}}-\mathrm{E}})} \\ ({10}^{-9}\mathrm{m})=\frac{6.63\times {10}^{-34}}{2\mathrm{\pi }\sqrt{2\times 9.1\times {10}^{-31}\mathrm{kg}({\mathrm{U}}_0-\mathrm{E})}} \\ ({\mathrm{U}}_0-\mathrm{E})=\frac{{(6.63\times {10}^{-34})}^2}{4{\mathrm{\pi }}^2\times 2\times 9.1\times {10}^{-31}\times {10}^{-18}} \\ ({\mathrm{U}}_0-\mathrm{E})=0.38\mathrm{eV} \end{gathered}$$