Prove that electron cannot reside inside the nucleus of an atom.

Using Heisenberg's uncertainty principle: \(\Delta x\,\Delta p\ge\dfrac{\hbar}{2}=\dfrac{h}{4\pi}\)

The diameter of a nucleus is about \(10^{-14}\) m. If an electron were confined inside it, the maximum uncertainty in its position would be \(\Delta x=10^{-14}\) m.

\[ \Delta p\ge\frac{h}{4\pi\Delta x}=\frac{6.63\times10^{-34}}{4\pi\times10^{-14}}=5.28\times10^{-21}\ \text{kg m/s} \]

The momentum itself must be at least of the order of its uncertainty, so \(p\approx5.28\times10^{-21}\) kg m/s.

Since \(pc\gg m_0c^2\) (\(pc\approx1.6\times10^{-12}\) J, while \(m_0c^2=8.2\times10^{-14}\) J), the electron would be relativistic, so \(E\approx pc\):

\[ E\approx pc=5.28\times10^{-21}\times3\times10^8=1.58\times10^{-12}\ \text{J}=\frac{1.58\times10^{-12}}{1.6\times10^{-13}}\ \text{MeV}\approx9.9\ \text{MeV} \]

(Using \(\Delta x\Delta p\ge\hbar\) instead gives about 20 MeV. The conclusion is the same.)

Conclusion: An electron inside the nucleus would need an energy of about 10 to 20 MeV. Experiments show that electrons emitted from nuclei (β-particles) have energies of only about 2 to 4 MeV, and electrons in atoms have energies of only a few eV to keV. Hence electrons cannot reside inside the nucleus.