I've been trying to find an answer to this question, but have come across contradictory answers, and have limited knowledge of quantum mechanics myself.
Almost all the threads (here and on Quora) state that the uncertainty principle prohibits the electron from entering the nucleus, for it would be required to travel with speeds greater than that of light, even though it has mass. However it is quite confusing, given that many support the probability of it being in the nucleus is not zero. Even if that is not true, wouldn't tunneling enable it to do it anyway?
Also when it enters, does it get captured by a proton to form a neutron as in a neutron star? If not what are the conditions required for this to happen?
I'd like to point out that this isn't a duplicate of Why don't electrons crash into the nuclei they "orbit"? as I am already aware of the quantum model, orbitals and principles like uncertainty and Pauli exclusion, and haven't been able to find a conclusive answer no matter where I've looked.
Its also different than What happens to Protons and Electrons when a Neutron star forms? although it's been a bit helpful (I think), but I am more interested in whether electrons and protons fuse into neutrons as soon as they get really close or there are more universal conditions than need to be satisfied before that happens (which obviously are in the specific case of neutron stars).
As Fermi brilliantly put it "Before I came here I was confused about this subject. Having listened to your lecture I am still confused. But on a higher level."
Edit: I wanted to get a bit more specific about the inconsistency I found. "In the quantum mechanical model of the electron, there is a finite probability of finding the electron within the nucleus. During the internal conversion process, the wavefunction of an inner shell electron (usually an s electron) is said to penetrate the volume of the atomic nucleus." Taken from the wikipedia page on internal conversion. Doesn't that come in conflict with the uncertainty principle based argument that the electron would have to travel faster than light?
Edit 2: To better explain my confusion I found two threads that summarize the arguments I find contradictory (although that could possibly be my flawed perception).
https://www.quora.com/Why-do-electrons-not-fall-into-the-nucleus-of-an-atom This is a quora thread about basically the same issue. The second answer by Rod Sid is a common explanation for why this phenomenon doesn't occur.
Yet according to Why do electrons occupy the space around nuclei, and not collide with them? it does. I hope you can see how these two can confuse people. Thanks for anyone that answers it in advance.