Question

| FIGURE EX$40.4$ shows the wave function of an electron in a rigid box. The electron energy islocalid="1650137157775" $12.0eV$. What is the energy, in localid="1650137162096" $eV$, of the next higher state?

Short answer

The energy, in $eV$is$21.3eV$.

Step-by-step solution

Given Information

We have to given The electron energy is$12eV$.

We need to find the energy in $eV$.

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 two maxima and one minimum, meaning that it has 3 antinodes. Therefore, the wave function shown in the graph corresponds to $n=3$The energy of an electron in a rigid box is given by:

$E_n=n^2{E}_1$

all need to do is to use the given value of $E_3$to calculate $E_1$, then use the equation above to determine the energy of the next state is $E_4$.

$\begin{array}{c}{E}_3=9E_1\\ E_1=\frac{E_3}{9}=\frac{12\text{eV}}{9}=\frac{4}{3}\text{eV}\end{array}$

So,

$\begin{array}{c}{E}_4=16E_1=16\times \frac{4}{3}\text{eV}\\ E_4=21.3\text{eV}\end{array}$