Question

Find the width L of a one-dimensional box for which the ground state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom.

Short answer

The width L of a one-dimensional box is$1.66\times {10}^{-10}m$ for which the ground state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom.

Step-by-step solution

Define the energy level.

The energy level of a particle of mass in a box width is:

$$\begin{gathered} E_n=\frac{P_n^2}{2m} \\ =\frac{n^2{h}^2}{8m{L}^2} \\ =\frac{n^2{\mathrm{\pi }}^2{\mathrm{h}}^2}{2m{L}^2}\left(n=1,2,3,...\right) \end{gathered}$$

Determine the width of box.

The energy$E_1$ of hydrogen at ground level is 13.6 eV .

The energy $E_1$in joule J is

$$\begin{gathered} E_1=13.6\times \left(\frac{1.60\times {10}^{-19}}{1}\right) \\ =2.18\times {10}^{-18}J \end{gathered}$$

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

The width of the box is:

$$\begin{gathered} L=\frac{n\mathrm{\pi h}}{\sqrt{2m{E}_n}} \\ =\frac{\left(1\right)\mathrm{\pi }\left(1.055\times {10}^{-34}\right)}{\sqrt{2\left(9.11\times {10}^{-31}\right)\left(2.81\times {10}^{-18}\right)}} \\ =1.66\times {10}^{-10}m \end{gathered}$$

Hence, the width L of a one-dimensional box is$1.66\times {10}^{-10}m$ .