Why don't electrons stick to the nucleus on account of being oppositely charged charged (the nucleus being positively charged)? But contrary to my intuition, electrons seem to follow a whole bunch of wacky paths (orbitals) around the nucleus (as demarcated by orbital wave-functions). Why is this?

Do note:

I'm not asking "What keeps the electron in orbit". All I want to know, is why electrons don't simply find themselves attached to the nucleus, and why do they even bother moving around the nucleus (in orbitals) at all? My textbooks, unfortunately, make no reference to this issue...they wholeheartedly adopt the "current" QM model of the (hydrogen) atom.

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    $\begingroup$ The cheeky answer to this is "because something keeps the electron in orbit around the nucleus." $\endgroup$ – chipbuster May 6 '17 at 21:19
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    $\begingroup$ This isn't about chemistry at all. Also I wonder what answer you'd even expect? $\endgroup$ – Mithoron May 6 '17 at 21:38
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    $\begingroup$ @Mithoron How isn't this about Chemistry? Surely, questions pertaining to the development and acceptance of the current model of the atom fall under the domain of Chemistry ( "Atomic Models" fall under both Chemistry and Physics...then again, the two subjects don't make a clear distinction in that regard). Going by your logic, this question would be unsuitable for even the Physics Stack Exchange. I genuinely appreciate that you voiced your opinion here, and I hope you can see this from my point of view as well. Thanks! $\endgroup$ – user44907 May 6 '17 at 21:49
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    $\begingroup$ What current model? Quite a bit of guys would even tell that they don't orbit at all! In chemistry orbits have almost no importance, only orbitals are used. $\endgroup$ – Mithoron May 6 '17 at 21:54
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    $\begingroup$ In a way, electrons do stick to the nucleus to form what we call an atom. There are no paths around. $\endgroup$ – Ivan Neretin May 6 '17 at 23:14

Richard Feynman alludes to this exact problem in the first volume of the Feynman Lectures on Physics. I'll present his argument here (based on the Uncertainty Principle), albeit, in my own words ;)

The German physicist [since he dealt with atomic chemistry/physics...by all means, go ahead and call him a "chemist" ;) ], Werner Karl Heisenberg came up with what we now know as "The Heisenberg Uncertainty Principle"... a revolutionary idea that points to an "inherent fuzziness", that exists within quantum systems (which is a fancy word that refers to any region of space that is sufficiently "small" enough to introduce wave-particle duality), and becomes apparent when we try to "measure" the various parameters that constitute them.

Basically, what (one version of) the Uncertainty Principle states is that:

The product of the uncertainty in measurement of velocity and the uncertainty in measurement of the position of a particle can never be less than a certain constant; i.e- ħ/2

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Rephrasing this

If we were to know a particle's position to a very high precision (i.e- small uncertainty in measurement of position) then the corresponding uncertainty in the measurement of the particle's velocity (or momentum, if you know its mass) increases greatly. The same holds true vice versa.

In other words,

You cannot know both a particle's position and momentum in a quantum system to a great precision simultaneously.

Here's a good analogy for this (Part 3: Atomic Structure - Analogy with a ballon)

From this point on, I guess the question can best be dealt with by "proof by contradiction".

First part of the question: Why don't electrons stick to the nucleus?

We know for a fact, that Heisenberg's Uncertainty Principle holds true (yeah, I didn't prove it...but for now, just take my word for it).

So, let's assume that electrons do stick to the nuclei of their respective atoms. In that event, we would know the electrons' positions and momenta (with respect to the nucleus) with almost absolute certainty...something that Heisenberg's Uncertainty Principle does not allow.

Seeing that this situation is impossible, it is obvious that our little "assumption" must be wrong (Knowing the the Uncertainty principle is indeed, correct).

So yeah, electrons don't just go and simply "stick" to atoms.

Second part of the question: Why do electrons follow a whole bunch of wacky paths?

The Dane, Niels Bohr postulated that electrons move in certain, fixed orbits around a nucleus...a fairly convincing (but not very accurate) model that is still often brought up in high-school and under-grad physics and chemistry courses. The Bohr Model was developed before the Uncertainty Principle.

But electrons do not move in fixed orbits ...the Uncertainty Principle triumphs again!

So if electrons don't stick to the nucleus, and if they don't move in fixed orbits around the nucleus, then where exactly are they?

We can never exactly point out an electron's position in an atom. However, we can determine the probability of finding electrons in a given region of space around the nucleus. These regions of space with a very high electron probability density are called orbitals.

This answer has been greatly over-simplified. Thought it does deal with the gist of the topic, I'm personally looking forward to seeing more detailed answers myself.

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    $\begingroup$ I wouldn't say it's oversimplified, but incorrect. Uncertainty Principle has no importance here. $\endgroup$ – Mithoron May 6 '17 at 21:47
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    $\begingroup$ @Mithoron Eh? But I recall hearing Feynman using the same argument: feynmanlectures.caltech.edu/I_02.html (Read page 2-3, Quantum Mechanics, 2nd paragraph). I think I'll trust Feynman on this one. Thanks for the down-vote by the way ;) $\endgroup$ – paracetamol May 6 '17 at 22:05
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    $\begingroup$ Electron degeneracy pressure not Uncertainty Principle. $\endgroup$ – Mithoron May 6 '17 at 22:49
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    $\begingroup$ The electron degeneracy argument is more to do with pushing atoms together and then what happens to energy levels via the Pauli Principle than with an isolated atom and its electrons and nucleus. Feynman's argument seem sound to me. $\endgroup$ – porphyrin May 7 '17 at 9:51
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    $\begingroup$ @Mithoron 1) electron doesn't fall into the nucleus even in a hydrogen atom, where degeneracy pressure doesn't exist, 2) degeneracy pressure comes from the uncertainty principle, as the very link you give also briefly mentions $\endgroup$ – Greg May 11 '17 at 17:23

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