I have an exam the day after tomorrow on the course "Nuclear Physics-I". While studying, I face the question, "why an electron cannot exist inside the nucleus?".

Have you ever thought about this? If yes, then congratulations! You just broke classical mechanics. This question is one of the fundamental questions that led to the formulation of quantum mechanics.

Okay, let me first show you what the book said about this question. So, it goes like this.

From Heisenberg's uncertainty principle, $$\begin{align*} & \Delta x \cdot \Delta p > \hbar \\ \Rightarrow & \Delta x \cdot m \Delta v > \frac{h}{4 \pi} \\ \Rightarrow & \Delta v > \frac{h}{4 \pi m \Delta x} \\ \Rightarrow & \Delta v > \frac{6.626 \times 10^{-34}}{4 \pi \times 9.10938 \times 10^{-31} \times 10^{-15}} \\ \Rightarrow & \Delta v > 5.79 \times 10^{10} \end{align*}$$ Does this look odd to you? The velocity of an electron will be greater than the speed of light if it goes within the radius of the nucleus. That's impossible.

While the math looked amazing to me, I wasn't really satisfied. Because, the math considered the electron to be a particle. But physics is now much fancier and we know electrons are particles only for certain cases. For other cases, it acts like a wave.

So now we get into the Fancy stuff. Actually, the electrons in an atom do enter the nucleus. Electrons in the p-orbital peak through the nucleus. Remember the dumbbell shaped orbital from chemistry classes? Yeah, that one. The state of the electron overlap with the nucleus.

Assuming you have studied quantum mechanics and now you are smart and might just say, "Fine! Electron is wave function. But is it possible to localise the electron in the nucleus?".

Then, me, who searched the internet already for this answer, say, "It is possible to localise the electron in the nucleus."

The process for that to happen is called "electron capture". This process is an important mode of radioactive decay. In this process, a proton in the nucleus absorbs an electron and turns into a neutron. Electron starts as a regular electron with its wavefunction overlapping with the nucleus. With enough time, the electron reacts with the proton by its overlapping part and collapses to a point and become a part of the nucleus. And now the nucleus has more neutrons and less protons. This is a type of radioactive decay. Because, the nucleus is now of another type of atom.