Question

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1. What is the length of the box?

Short answer

Tthe length of the box is $L=426.4\mathrm{pm}$

Step-by-step solution

Given information

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1.

Explanation

Energy levels of the particle in the box:

$E_n=\frac{h^2{n}^2}{8m{L}^2}$

Energy of the photon:

$\Delta E_{12}=\frac{hc}{\lambda }$

Substitute energy levels:

role="math" localid="1648514586435" $$\begin{gathered} E_2-E_1=\frac{h^2·2^2}{8m{L}^2}-\frac{h^2·1^2}{8m{L}^2} \\ {E}_2-E_1=\frac{3h^2}{8m{L}^2} \end{gathered}$$

Substitute energy of the photon:

$\frac{hc}{\lambda }=\frac{3h^2}{8m{L}^2}$

Calculations:

Express the above equation in terms of L:

$L=\sqrt{\frac{3h^2\lambda }{8mhc}}$

Substitute the values:

$$\begin{gathered} L=\sqrt{\frac{3{\left(6.63·{10}^{-34}\right)}^2·200·{10}^{-9}}{8·9.11·{10}^{-31}·6.63·{10}^{-34}·3·{10}^8}} \\ L=426.4\mathrm{pm} \end{gathered}$$