Question

The charge on a muon-a subatomic particle is -e and its mass 207 times that of an electron-

which is confined in a 15-pm-long, one-dimensional box.

(1 pm = 1 picometer = 10^-12 m.)

What is the wavelength, in nm, of the photon emitted in a quantum jump from n=2 to n=1

Short answer

The wavelength is $\lambda =51.2\mathrm{nm}$

Step-by-step solution

Given information

Mass of muon-a subatomic particle is 207 times that of an electron-is confined in a 15-pm-long, one-dimensional box.

Its charge is -e

(1 pm = 1 picometer = 10^-12 m.)

Explanation

The energy levels of the particle in the box is:

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

Energy of the photon is:

$\Delta E_{12}=E_2-E_1$

Substitute the energy levels:

role="math" localid="1648510290213" $$\begin{gathered} E_2-E_1=\frac{h^2·2^2}{8m{L}^2}-\frac{h^2·1^2}{8m{L}^2} \\ {E}_2-E_1=\frac{3h^2}{8m{L}^2} \end{gathered}$$

Energy of the photon is:

$E_2-E_1=\frac{hc}{\lambda }$

Substitute energy of the photon:

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

Calculations

Expressing the equation for $\lambda$

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

Substituting the values:

$$\begin{gathered} \lambda =\frac{8·6.63·{10}^{-34}·3·{10}^8·207·9.11·{10}^{-31}·{\left(15·{10}^{-12}\right)}^2}{3·{\left(6.63·{10}^{-34}\right)}^2} \\ \lambda =51.2\mathrm{nm} \end{gathered}$$