Could an electron enter the nucleus, and if so would it be captured by a proton? 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.
 A: You may want to take a closer look to electron capture phenomenon, also in this thesis.
A: Actually, a similar question was asked by Feynmans dad to Feynman after he graduated from MIT and Feynman found it difficult to answer - which shows by the way, that it is a good quetion.
As Feynman pointed out, the problem lies with the premises of the question which presupposes things like entering or leaving. These are notions common to a purely material atomic theory, where atoms enter a new configuration or leave one. 
In fact, what we have is identity change. 
Whilst a neutron may decay into a proton and an electron, in no sense is a neutron ontologically speaking the same as a proton and an electron. The decay describes a process of ontological change. The neutron changes into a proton and an electron. 
You might find it interesting that this one of the reasons that Aristotle dismissed the atomic theory, as he regarded such ontological changes as possible, and in fact neccessary, but the atomists merely saw change as a kind of permutation of matter; and this over two millenia ago!  
A: Electrons in the atom do enter the nucleus. In fact, electrons in the s states tend to peak at the nucleus. Refer to https://www.wtamu.edu/~cbaird/sq/2013/08/08/why-dont-electrons-in-the-atom-enter-the-nucleus/
A: All the references you refer to use a sort of semi-classical semi-quantum language, and you yourself admit you are confused about quantum mechanics. So it might be helpful to have a 'pure' quantum answer (although it might not directly address your question):
Quantum particles do not possess a property called 'position'.
This sounds strange, since you can measure where the particle is (eg by constructing a box that tells you if there is a particle inside of it, then waiting until it beeps to signify the particle is in the box). Measurement is a complicated (and I think poorly understood?) process where an object so complicated that you need to resort to a classical system (the measuring device) interacts with a system you are trying to treat quantum mechanically. The result is tainted by the classical measurement device and you end up with classical notions like position in your output. 
So the simple answer to the question is that no, an electron cannot be in the nucleus, because it cannot be anywhere. It does have a finite chance to be found in the nucleus if you were to measure it (again, imagine putting a box around the nucleus that tells you if there is an electron there, there is a finite chance it says yes). The fact that electron capture occurs means that the full quantum solution means there is a chance that the atom will later be found in a state which describes one less proton and electron and one more neutron. The semi-classical interpretation is that the electron spends a small fraction of its time in the nucleus (corresponding to whatever the probability of your magic measuring box saying it is there) and so there is a chance to "tunnel" in to the nucleus, fuse with an attractive proton and create a neutron. These semi-classical pictures can give accurate answers but they are not the full quantum picture.
Edit 1: Uncertainty Principle
The argument with the uncertainty principle is I believe being misapplied. Leaving aside any issues of interpretation, the uncertainty principle argument runs that if you locate the particle to be in the nucleus, then you must only know its momentum to some uncertainty:
$$\Delta x \Delta p \sim h$$
Where $h$ is Planck's constant and $\Delta x$ is the uncertainty in our position measurement. If we are claiming to have measured the electron to be "in the nucleus" then we must have $\Delta x$ roughly the radius of the nucleus. (It's been pointed out in the comments that the notion of radius is not commensurate with positions not being quantum mechanical. That's okay, we're treating the nucleus classically here anyway.) This forces $\Delta p \sim \frac{h}{\Delta x}$ and so the uncertainty in the particle's momentum becomes infinitely large as we try to confine the particle to a given location.
There are two things to comment on here: first there is the rather boring statement that the uncertainty in the momentum is not actually the momentum itself. Clearly if $\Delta p$ is very large we expect $p$ to have a probability distribution peaked around some large value that is at least $\Delta p$ but it doesn't preclude $p=0$ directly. Secondly (and more importantly), the mechanism for stopping electrons "crashing into the nucleus" is that as their momentum increases due to the uncertainty principle, they spend less time in the nucleus and the amount of time they do spend is not enough to fuse with the nucleus (i.e. they go through it too fast to fuse usually). It is NOT that the electron starts moving faster than the speed of light $c$ because the usual formula $p=mv$ stops working at relativistic speeds in such a way that the speed remains bounded below $c$ even as $p$ becomes arbitrarily large. 
A: In general electron capture does not occur because the final state has much higher energy than the energy that the electron contributes. The simplest case is $^1H$ or the proton. The neutron rest energy is $1.3~MeV/c^2$ larger than that of the proton. Of course, then the question is why does it ? This brings us to nuclear physics and the quark model. The neutron composition is $uud$ and the proton has $udd$. Quark rest energies do not explain this. $d$ is actually $2.5~MeV/c^2$ heavier than $u$. Most of the nucleon rest energy comes from the gluons and so does the n-p difference. For some nuclei the energy actually goes down upon transforming a p into a n so that electron capture can occur.
Then why, in hydrogen, does the electron not go closer to the nucleus than 1 angstrom? The reason is that in quantum mechanics the kinetic energy is proportional to the squared gradient of the orbital and increases faster than its potential energy decreases when it is shrunk.
