91Ó°ÊÓ

The given data is listed below as-

The length of the narrow-evacuated tube L = 3m

Theenergies differenceis given by the equation-

$\mathbf{∆}\mathit{E}\mathbf{=}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}\mathbf{m}{\mathbf{L}}^{\mathbf{2}}}\left(n_2^2-n_1^2\right)$

Here, L is the length of the tube.

The difference between energies is given by the equation:

$∆E=\frac{h^2}{8m{L}^2}\left(n_2^2-n_1^2\right)$

For, $n_2=2$and $n_1=1$

$$\begin{gathered} ∆E=\frac{\left(6.626×{10}^{-34}\right)}{8×9×109×{10}^{-31}×3^2}\left(2^2-1^2\right) \\ =2×{10}^{-38}J \\ =1.3×{10}^{-19}eV \end{gathered}$$

Thus,the energy difference between the electron’s ground state and its first excited state is$=1.3×{10}^{-19}eV$ .

The difference between energies is given by equation:

$∆E=\frac{h^2}{8m{L}^2}\left[{\left(n+1\right)}^2-n^2\right]$

The value of n is obtained by solving the above equation as below:

$n=\frac{4∆m{L}^2}{h^2}-\frac{1}{2}$

Substituting all the values in above equation:

$$\begin{gathered} n=\frac{4×1.602×{10}^{-19}×9.109×{10}^{-31}×3^2}{\left(6.626×{10}^{-34}\right)}-\frac{1}{2} \\ n=1.2×{10}^{19} \end{gathered}$$

Thus, at quantum number $n=1.2×{10}^{19}$ the energy difference between adjacent energy levels will be 1.0 ev-which is measurable.

At$n=1.2×{10}^{19}$,the energy equation is given as

$$\begin{gathered} ∆E=\left(\frac{h^2}{8m{L}^2}\right)n^2 \\ =\frac{{\left(6.626×{10}^{-34}\right)}^2}{8×9.109×{10}^{-31}×3^2}\left(1.2×{10}^{19}\right) \\ =0.96J \end{gathered}$$

The rest energy is given as:

$$\begin{gathered} E_{rest}=m{c}^2 \\ =9.109×{10}^{-31}×3×{10}^8 \\ =8.2×{10}^{-14}J \end{gathered}$$

Now, to find multiple of the electron’s rest energy that would give the electron’s total energy, divide the value of E by $E_{rest}$.

$$\begin{gathered} \frac{E}{E_{rest}}=\frac{1.13}{8.2×{10}^{-14}} \\ =1.2×{10}^{13} \end{gathered}$$

Yes, the electron is relativistic because the kinetic energy of this electron would give the nonrelativistic equation which suggests that its speed is greater than the speed of light.