Question

Three electrons are in a one-dimensional rigid box (i.e., an infinite potential well) of length $0.50nm$. Two are in the $n=1$state and one is in the $n=6$state. The selection rule for the rigid box allows only those transitions for which $∆n$is odd.

a. Draw an energy-level diagram. On it, show the filled levels and show all transitions that could emit a photon.

b. What are all the possible wavelengths that could be emitted by this system?

Short answer

For one dimensional rigid box the energy levels are as follows.

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

Length of the box is,

role="math" localid="1648742877385" $=1.51n^{2}eV$

Step-by-step solution

For one dimensional rigid box the energy levels are as follows.

$$\begin{gathered} E_n=n^2\frac{h^2}{8m{l}^2} \\ Lengthoftheboxis, \\ L=0.5nm \\ =(0.5nm)\left({10}^{-9}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$nm$}\right.\right) \\ =0.5x{10}^{-9}m \\ Substituting6.63x{10}^{-34}j.sforh9.1x{10}^{-24}kgform.0.5x{10}^{-9}mforL \\ {E}_n=n^2\left(\begin{array}{c}{\left(6.63x{10}^{-34}js\right)}^2\\ \left(8\right)\left(9.1x{10}^{-31}kg\right){\left(0.5x{10}^{-9}m\right)}^2\end{array}\right) \\ =\left(2.415x{10}^{-19}j\right)n^2 \\ =\frac{\left(2.415x{10}^{-19}j\right)n^2}{\left(1.6x{10}^{-19}{\displaystyle \raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$eV$}\right.}\right)} \\ =1.51n^{2}eV \end{gathered}$$