quantum mechanics - Why can't electrons be found inside the nucleus if there are infinite number of orbitals?

If there are an infinite number of orbitals, we can assume, that they can be present in any point in space. If that is correct, why do we not find electrons in the nucleus?

I study in high school. Correct me if I'm wrong.

Answer

Let's suppose the electron we are considering is in an orbital described by the wavefunction $\psi$. If we look in some small volume element $dV$ then the probability of finding the electron in that volume element is:

$$ P = \psi^*\psi \, dV $$

To calculate the probability of findng the electron inside the nucleus we'll use polar coordinates, and as our volume element $dV$ we'll take the volume of a spherical shell of radius $r$ and width $dr$. The volume of this element is:

$$ dV = 4\pi r^2 dr $$

so the probability is:

$$ P = \psi^*\psi \, 4\pi r^2 dr $$

If the radius of the nucleus is $R$, then we get the probability of finding the electron in the nucleus simply by integrating from $r = 0$ to $r = R$:

$$ P = \int_0^R \psi^*\psi \, 4\pi r^2 dr $$

And this integral generally has a non-zero magnitude i.e. the probability of finding the electron inside the nucleus is non-zero.

We know the electron has a non-zero probability of being inside the nucleus because in some cases it can react with a proton in a process called electron capture or inverse beta decay.