91Ó°ÊÓ

Given that the radius of the atomic nucleus is of the order of $5×{10}^{-15}m$. We can take the uncertainty in position of the proton inside the nucleus, $∆r=5×{10}^{-15}m$.

From the Heisenberg’s Uncertainty Principal for momentum and position,$\mathbf{∆}{\mathit{p}}_{\mathbf{r}}\mathbf{∆}\mathit{r}\mathbf{â‰\yen }\frac{\mathbf{h}}{\mathbf{2}}$ .

The uncertainty in momentum will be

$$\begin{gathered} ∆p_râ‰\yen \frac{\sout{h}}{2∆r} \\ ∆p_râ‰\yen \frac{1.055×{10}^{-34}}{2×5×{10}^{-15}} \\ ∆p_râ‰\yen 1.05×{10}^{-20}kg.m/s \end{gathered}$$

So, the uncertainty in the momentum is $∆p_râ‰\yen 1.05×{10}^{-20}kg.m/s$.

Taking the minimum uncertainty in the momentum of proton as an estimated momentum, $p=1.05×{10}^{-20}kg.m/s$.

Now, solve for E.

$$\begin{gathered} E^2={\left[\left(1.05×{10}^{-20}\right)\left(3×{10}^8\right)\right]}^2+{\left[\left(9.11×{10}^{-31}\right)\left(3×{10}^8\right)\right]}^2 \\ {E}^2=1×{10}^{-23} \\ E=3.16×{10}^{-12}J \end{gathered}$$

Using the energy, we can find the value of .

$$\begin{gathered} K=E-m{c}^2 \\ K=3.16×{10}^{-12}-\left[\left(9.11×{10}^{-31}\right){\left(3×{10}^8\right)}^2\right] \\ K=3.08×{10}^{-12}J \end{gathered}$$

Therefore,$K=3.08×{10}^{-12}J=19.267eV$

We know that the Coulomb potential energy between two charges${\mathit{q}}_{\mathbf{1}}$ and ${\mathit{q}}_{\mathbf{2}}$separated by a distance r is given by $\mathit{U}\mathbf{=}{\mathit{k}}_{\mathbf{e}}\frac{{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{\mathbf{r}}$.

For a proton with charge$q_1=+e$and an electron with a charge$q_2=-e$and separated by a distance $r=5×{10}^{-15}m$, Coulomb potential energy is

$$\begin{gathered} U=8.988×{10}^9\frac{-{\left(1.6×{10}^{-19}\right)}^2}{5×{10}^{-15}} \\ U=-4.6×{10}^{-14}J \\ U=-0.288MeV \end{gathered}$$

As we can see the kinetic energy of an electron inside a nucleus as calculated by the uncertainty principle is larger than the coulomb potential energy.

So, an electron inside the nucleus can easily overcome the coulomb potential and leave the nucleus even with a very high kinetic energy.

Therefore, an electron can not remain in the nucleus.