Question

| FIGURE EX $40.3$shows the wave function of an electron in a rigid box. The electron energy is$25eV$. How long is the box?

Short answer

The electron energy is $25eV$ so the box is$0.736nm$.

Step-by-step solution

Given Information 

We have to given The electron energy is$25eV$.

Simplify

The graph of ${\psi }_n\left(x\right)$has $\left(n-1\right)$nodes, excluding the ends, and $\left(n\right)$antinodes. In our case, the wave function shown in the graph has three maxima and three minima, meaning that it has six antinodes. Since the wave function is shown in the graph corresponds to the $n=6$state. The energy of levels of an electron in a rigid box is given by:

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

Then,

$E_6=\frac{36h^2}{8m{L}^2}$

the equation to solve the numerical values of the different variables.

$$\begin{gathered} L=\frac{6h}{\sqrt{8m{E}_6}}=\frac{6\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)}{\sqrt{8\left(9.11\times {10}^{-31}\text{kg}\right)\left(25\text{eV}\times 1.6\times {10}^{-19}\text{J}/\text{eV}\right)}} \\ L=7.36\times {10}^{-10}\text{m}=0.736\text{nm} \end{gathered}$$