Electron falling into proton approaches infinite kinetic energy why?

Question

Foreword

I believe this question to be different from the usual "Why don't electrons fall into the nucleus" because this

A: asks about the details of explanations for why electrons don't fall into the nucleus.

B: comes from a different level of understanding about the electron models and doesn't start with a 'bohr' model.

C: assumes the explanations to be true and instead asks about Why they are true, Especially when talking about 'infinite' energies.

I have stackoverflow experience so I try to follow the rules best I can but this is my first question on this stack exchange variant so I apologize if I do anything wrong with regards to the specifics of physics stack exchange (I've tried reading the help page so thats what I'm going off of).

My current level of understanding is that I've taken Physics 1 and 2 college courses so I have a basic understanding of the basic newtonian laws + Electromagentism and some of the more modern stuff and I know a (tiny) bit about quantum effects. Wave particle duality is familiar to me but I don't understand why it is so, only that it is that way. So please if you have any equations I'd appreciate links and a reference to what the variables mean as I won't recognize them (unless its like e=mc^2 or something), But I am good at math.

What I'm doing/what I know

I'm trying to understand the explanation(s) for why electrons don't fall into the nucleus because my physics 2 professor wasn't able to provide an explanation. So far I understand that:

1. electrons are not balls of matter orbiting the nucleus like in the bohr modle, rather they are a probability cloud around the nucleus.

2. that electrons have a smaller mass and negative charge that when interacting with a positive charge of a nucleus causes an attraction force and that when this happens photons are emitted

3. The Heisenberg Uncertianty principle means that determining an electron's position and momentum at the same time is impossible and that when you know an electron's position you can't have any idea what momentum it has and vice versa.

The main question

The question I have is mainly about the details of the explanation(s) for why the electron doesn't 'fall into' the nucleus (even though I've heard it can spend non trivial amounts of time in the nucleus). So far I've seen it said that as the electron cloud approaches the nucleus it's kinetic energy approaches positive infinity and it's potential energy approaches negative infinity. Also, one of the provided explanations is that when the electron reaches the nucelus, it has a position which is almost certain, thus due to the Hiesenberg uncertianty principle, that means its momentum must be very uncertian and thus large. I have many questions about this explanation for why this cloud of electron probabilities has a limit of infinities (potential energy -infinity to kinetic energy +infinity) and why it is said that it converges to around a factor of 2 (with kinetic energy being greater thus causing a 'stable orbit' because the electron can escape but not too much).

I've also heard that the reason for electrons not falling into the nucleus is because they are already in the lowest energy state possible and thus are stable. I don't see where this stability is coming from unless its from the above balance between kinetic and potential energies because I can't see any reason for the nucleus to repel the electron 'cloud'.

How I've tried to envision it

From a simple energy perspective I can understand why a kinetic energy dominance would cause a stable orbit but the problem I have is seeing why the kinetic energy dominates in the first place, why the kinetic energy dominates by a factor of about 2, and why it approaches positive infinity and why the potential energy approaches negative infinity when the electron approaches the nucleus. The infinity part of this is the most confusing part to me because;

From my simplistic point of view, I have a finite distance from electron cloud to nucleus and I have two 'particles' with finite masses with finite energies and an interaction in finite time. If this is true then where does this 'infinite' kinetic energy come from?

I try to compare this to what I would imagine a more standard model of the 'motion' would be. For example, modeling a planet falling into a star with gravity as the force of interaction doesn't cause the planet to gain infinite kinetic energy because it can only ever have a finite potential energy because it is a finite distance away and has finite mass and falls inside the star in finite time. Sure the planet can escape but only if it has enough potential energy to begin with. It doesn't ever have infinite kinetic or potential energy. Even if you make the planet a 'cloud' with no determinable momentum AND position and change gravity to electromagnetic attrraction wouldn't the same basic laws of motion still apply? Yes I know quantum effects are weird with regard to motion and stuff but I want to know if/how/why its different in this case.

Is it truly Infinite?

The other thing that confuses me about these 'infinite' energies is that they are infinite, which means we should see some crazy things, like crazy amounts of heat or mass or light being created right? So how in the world does this work? Is there some other limit I'm not taking into account here?

Possible solutions

The only thing I could possibly envision as a solution to this problem is that

A(likely): I'm thinking about it entirely wrong OR

B: the kinetic energy doesn't actually become infinite and rather it is simply always large enough to enable an eventual escape of the electron cloud from the nucleus.

I could only see B coming from some sort of repulsive force from the nucleus. I can't however see where that force comes from, gravity wouldn't matter on that small a scale, I think the strong force would only attract the particles, the weak force I don't know much about, and the electromagnetic force is already the main force of attraction being discussed so where would any repulsion come in? I remember seeing somewhere that neutrons might repel electrons in some cases but then that leaves open the question of electrons falling into hydrogen nuclei with no extra neutron.

Maybe I'm reading it wrong and it isn't infinite

But if the energies wern't infinite then how can it be garenteed that electrons shouldn't always/almost always fall into the nucleus? Is it garenteed at all? Because according to my current understanding it seems to be garenteed at least somewhat because if it weren't it would break almost everything. Clearly that isn't happening since I can type right now and my atoms aren't just spontaneously collapsing in on themselves and becoming neutrons. Perhaps only a small number actually do this and that number is so small it becomes trivial? I don't really know but I thought that (at least for stable isotopes) there wouldn't be any decay or something like that but radioactive decay is totally different from an atom collapsing in on itself so I could be misunderstanding this.

Sidepoint: If electrons can and do spend non-trivial time inside the nucleus, does that change the behavior of the atom itself while the electron is inside the nucleus? Do they actually contact each other? Because supposedly thats supposed to create a nuetron and that would probably affect alot of things.

So, I humbly request some clarification on this issue. Why infinite energies? Why do they converge to two? Why does kinetic energy dominate here? Is any of this right at all? Do they actually fall into each other and create neutrons? Do they never do this and its garenteed to be so?

Answer 1

You are stuck in the classical framework. It is an experimental fact that once the existence of electrons and nuclei was established by observations, and atomic spectra were observed, no classical electromagnetic solution could fit the data. So the answer to why the electron does not fall on the proton and reach the infinite energies that the 1/r classical potential predicts is: because that is what has been experimentally observed.

Atomic spectra , together with the photoelectric effect and black body radiation necessitated the introduction of a new mathematical theory that can fit the data and predict generally the behavior of atoms and molecules and much more. It is called quantum mechanics.

There are no orbits or trajectories for particles at the microlevel of electrons and nuclei, there are only probabiliy distributions, called orbitals, and the solutions of the equations that give these orbitals do not allow reaching the r=0 of the 1/r potential in the classical sense. There is a probability for an electron to pass through r=0 but that is all. There is no 1/r there. It has been used to calculate the orbitals.

I have a number of links in my answer to a similar question here.

Edit after comments:

look at the evaluation of the electron orbitals for the hydrogen atom in the x,y plane.

The s orbitals pass through r=0. ( zero angular momentum quantum number) At the same time the corresponding orbital of the proton ( let us not forget the nucleus) will also look like this but for a much, orders of magnitude smaller dimension. The combined probability of overlap is smalland the energy of the s state is not enough to make a neutron. Thus in the case of the electron and the proton this can happen in neutron star conditions, energy supplied by the gravitational field. Electron capture does happen with heavy nuclei where there is the available energy to change a proton into a neutron, with quantum mechanical evaluated probabilities.

Answer 2

Classically it's the same reason the earth doesn't fall into the sun or the moon into the earth. The equation of motion of a mass attracted to a central attractor is an orbit. So pre-quantum mechanics, physicists wouldn't be surprised by the model of electrons orbiting a positively charged nucleus. Now even suppose that sometimes an electron did crash into a nucleus and change a proton to a neutron. How would you know it had happened? In the case of hydrogen, the neutron would disappear, you'd have one atom less, and only hydrogen stable orbits would remain. So there is almost an anthropic principle there: Hydrogen is stable and what's not stable is not hydrogen! But now comes quantum theory, and it had better predict the already observed behavior, hopefully even more accurately, otherwise the theory would not have survived.

Answer 3

The simplest answer you probably know, but will reiterate: electrons are not the Green functions of their field, they are the field, hence they evolve according to a wave-like equation, the stable ground solutions exist for the case with central potentials, and those are well known. The idea of infinite energy only happens when you try to extrapolate the point-like model inside the Compton radius

Answer 4

Lets start with the things you have got totally right, then move to the resolution of your concerns.

1. According to 19th century physics, the way an object falls into planet under gravity is exactly the same mathematically as the way an electron falls into an atom under the electric force. (assuming both start at rest).

2. If you assume a point source (i.e. a point size planet or point size atomic nucleus) then you can do a calculation of motion before the object/electron reaches the centre. You find that the object reaches the centre in a finite time, but during this time it reaches infinite velocity so has infinite kinetic energy. This doesn't contract Newton's Laws - as you reach the centre the force gets infinitely large so (F = ma) acceleration becomes infinite so it is quite reasonable that velocity becomes infinite in a finite time.

3. Now for the resolution we need to break it up into the two case. First gravity: here the resolution is clear: any planet has its mass spread over a region so the object will hit the surface of the planet before we reach any absurd infinities.

4. But now to the atom. Here things are much more tricky. The nucleus of an atom is more or less point like and in terms of 19th century physics there is nothing to stop an electron crashing into it at immense speed.

5. The resolution of this, I am afraid, cannot come in the language of Classical Physics we are using here. To solve the problem required the largest revolution in the history of physics: Quantum Mechanics. No short cut other than learn about that.

Answer 5

If your problem is about the word "infinite", the answer is simple.

What is meant is that *if you were to try to get closer and closer to the nucleus, at a distance $r$ increasingly small, the potentiel energy will become increasingly large, with no upper (or rather, since it is negative, no lower) limit. It goes like $-A/r$ with $A$ a constant the exact value of which is not important for my argument. As long as $r$ is not zero, it is finite but as you see there is no upper limit to the absolute value. This is the meaning you should understand about the word "infinite"

If you try to confine an electron into a volume of increasingly small size $d$, quantum mechanics proves that the kinetic energy of the particle increases indefinitely. Why this is so needs a deep understanding of quantum mechanics which I don't want to go into.

I am only addressing your concern about "infinite".

The point is that the positive kinetic energy increases as $B/d^2$. Note that this is not kinetic energy of motion towards the nucleus. You have to think of it a "internal" kinetic energy. A comparison that is far from being adequate but can give you a feeling, is like the internal energy of a gas that one compresses into a smaller volume. This internal energy is the sum of the kinetic energy of each molecule, not kinetic energy of global motion of the gas itself.

$A$ is a classical quantity. $B$ is a purely quantum one and thus is extremely small in "classical" situation. Consider an electron allowed to fall towards a proton from a "classical" distance. You understadn that from a QM perspective, the electron is a "cloud", but as long as the lectron is far from the nucleus, the cloud need not be very tight, and all of it "feels" the negative potential energy $-A/r$ which increases more and more i nabsolute value . Because of energy conservation this "deepening" negative energy is compensated by an increasing kinetic energy. As long as the electron is not too tight, $d$ not too small, because $B$ is of quantum origin, the "internal" energy is negligible.

But it is impossible for the electron to "feel" the indefinitely increasing negative potential energy $-A/r$ unless $d$ becomes of the same size as $r$.

For $r$ small enough, both $-A/r$ and $B/d^2$ for $d$ about the size of $r$ keep increasing indefinitely as $r$ decreases. But the second one increases as the square of $d \approx r$, much faster a $r$ decreases.

However small $B$ is because it is quantic, for small enough (but still finite) $r$, $B/d^2$ with $d \approx r$ will be equal to the contribution of the potential energy. All the kinetic energy will become "internal", compression kinetic energy of the smaller size of the electron cloud.

And thus the electron cannot get closer to the nucleus, because the expenditure in "internal", compression kinetic energy cannot be "afforded" by the negative potential energy.

This is a purely qualitative description. As you see, nothing is infinite.

The "infiity" business is just to mean that, however small the quantic coefficient $B$ is, for small enough $r$, this term will dominate because it "goes towards infinity" as $B/r^2$ faster than the potential energy, $-A/r$.