Why Don't Electrons Just Crash Into the Nucleus?

Here's a question that may have come to your mind in the past: why don't electrons just crash into the nucleus? 

Of course, we've been taught (since middle school science!) that electrons orbit the nucleus just because. However, they also teach us that opposite charges attract. Seeing that electrons are negatively charged and protons are positively charged, we have to question how the electron is able to maintain its orbit. Why doesn't it get slowly drawn towards the positive nucleus? 

A common (and incorrect!) explanation is that it is the speed of the electrons that keep them from flying toward the nucleus. In other words, the electrons are zooming around so fast, that they are able to overcome the attractive force. This idea is similar to spinning keys on a string; even though the string is pulling the keys towards your hand, the speed of the keys keep it moving in a circle. 

This explanation would be just dandy if not for the fact that, if you were to run the calculations, electrons would have to be moving faster than the speed of light to achieve this feat! That's obviously impossible…so, on to the next explanation…that is unfortunately, a tad bit more complicated! 

The very first thing you need to do is get the following image of electrons out of your head: 

Sorry to burst any bubbles, but this image of an electron as some sort of tiny, hard sphere is simply wrong. We have to use our imaginations a bit in our visualization of electrons so that we can incorporate quantum mechanics into this problem. Under quantum mechanics, an electron is more of a "cloud"; instead of being a tiny ball existing in one spot, it is "spread out" over the volume of the atom. And it's not even a physical hazy mist kind of cloud - it's more of a cloud of probabilities. The denser this cloud, the more likely the electron is to be there, as seen in this diagram: 

The dense ring around the nucleus is where the electron is most likely. Of course, you may have taken note of our use of the word  "likely" - that's right, we actually cannot be sure exactly where the electron will be in the atom! We can only speak in terms of probability. As you can see in the above diagram, there are some dots way on the outskirts of the image and there are definitely a few dots right up against (or even inside of) the nucleus. Even though there is a chance that the electron could be inside the nucleus or way, way outside of the atom, it is extremely unlikely. I mean, it's difficult to even express how ridiculously unlikely it is. 

Uncertainty 

As stated above, the electron's position in an atom is fundamentally uncertain. No matter what we do or what measurements we perform, we can never know, with exact precision, the position of an electron. For that matter, we actually can't even know how fast an electron is moving with exact certainty either! These two quantities are related by Heisenberg's Uncertainty Principle: 

                                                                    

Delta X and Delta P represent the uncertainty in a particle's position and momentum, respectively. The above is what we call an inequality; the left side must be more than the right side (the right side being an extremely small number). Upon some thought, it is clear that neither Delta X nor Delta P could ever be 0 - if one of them was, the entire left side would be 0 and that would violate this inequality and, consequentially, Heisenberg's Uncertainty Principle. This is explained further here.  

So, what does this have to do with electrons crashing into nuclei? Well, the thing is, now that we've established that electrons are more like clouds, we have to shift our thinking from "electrons crashing not nuclei" to "the electron cloud hitting the nucleus". And of course, there is a chance (a very minuscule chance) that an electron's cloud will hover over at east some portion of the nucleus. 

And here is why it is unlikely that an electron cloud will "hit" the nucleus: 

The nucleus is a very small spot in the atom. For an electron to actually be somewhere within the nucleus, it's cloud of probability would have to be very small (since it would have to hover mostly over the nucleus). From Heisenberg's Uncertainty Principle however, we know that as certainty in position decreases (i.e. the cloud shrinks), the momentum of the electron must increase proportionally. This means that, even if we were able to narrow in the electron to somewhere in the nucleus, its momentum would be extremely large, meaning it would be "zooming" around very fast! Alls this did was increase its uncertainty again because now its momentum has probably taken it somewhere else in the atom! 

So to reiterate, while it is possible for electrons to "crash" into a nucleus, it is extremely unlikely for the reasons described above.