Question

An electron is bound in a square well of depth${\mathbf{U}}_{\mathbf{0}}\mathbf{=}\mathbf{6}{\mathbf{E}}_{\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{DW}}$. What is the width of the well if its ground­ state energy is$\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{eV}$?

Short answer

The width of the well is$3.43\times {10}^{-10}\mathrm{m}$, if its ground state energy is$2.00\mathrm{eV}$.

Step-by-step solution

Define the energy level.

The energy levels of a particle of mass$\mathbf{m}$in a box width$\mathit{L}$is:

$$\begin{gathered} {\mathit{E}}_{\mathbf{n}}\mathbf{=}\frac{{\mathbf{P}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}\mathbf{m}{\mathbf{L}}^{\mathbf{2}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}{\mathbf{\pi }}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}{\mathbf{L}}^{\mathbf{2}}}\left(n=1,2,3...\right) \end{gathered}$$

The ground state energy as a fraction of the ground-level energy for an infinitely deep well$\left({\mathrm{E}}_{1-\mathrm{IDW}}\right)$is given by:

$E_1=0.625E_{1-1DW}$

Determine the width of the well.

Given that, the energy level of square well of depth is$U_0=6E_{1-IDW}$and ground state energy is$2.00\mathrm{eV}$.

The ground-state energy ${\mathrm{E}}_1$:

$$\begin{gathered} {\mathrm{E}}_1=2.00\times \left(\frac{1.60\times {10}^{-19}}{1}\right) \\ =3.60\times {10}^{-19} \end{gathered}$$

The mass of electron is $9.11\times {10}^{-31}$and value of $\mathrm{h}=1.055\times {10}^{-34}\mathrm{J}\cdot \mathrm{s}$.

The width of the well is:

$L=\sqrt{\frac{0.625{\mathrm{\pi }}^2{\mathrm{h}}^2}{2m{E}_1}}$

$$\begin{gathered} =\pi \left(1.055\times {10}^{-34}\right)\sqrt{\frac{0.625}{2\left(9.11\times {10}^{-31}\right)\left(3.20\times {10}^{-19}\right)}} \\ =3.43\times {10}^{-10}\mathrm{m} \end{gathered}$$

Hence, the width of the well is $3.43\times {10}^{-10}\mathrm{m}$, if its ground state energy is $2.00\mathrm{eV}$.