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Without going deep into mathematics and simply using symmetry arguments I made the following observations-

  1. An electron has a probability of being at a particular position
  2. Let's take the 1s orbital of a Hydrogen like single electron species and a uniform sphere at some distance $r$ from it.
  3. By Symmetry we can conclude the electron is equally likely to be present at any point on the sphere.
  4. Hence for any point P on the sphere there will be a diametrically opposite point Q passing through the nucleus where the electron is equally likely present.
  5. Let's say at 2 different times the electron is actually present there so for those instants the mass of electron is present at that point.
  6. All of this happens very quickly in fractions of seconds it can be on some point and then at another so for an observer it would seem it's present at multiple points right?
  7. If the observer then tries to calculate the average position of the electron taking symmetry arguments wouldn't it lie on the nucleus itself as the average of two diametrically opposite points is the midpoint of the diameter i.e. the nucleus.

With this isn't this somewhat similar to saying the Centre of Mass of the electron lies at the nucleus as the centre of mass is the point where we can assume the mass to be concentrated and we can take similar symmetry methods to calculate say the centre of mass of a uniform solid sphere to be at the centre.

This is just something I thought of a while ago, does this make sense to conclude?

Also if I were to extend this to multi electron species how would the reasoning go? How would I incorporate the effects of other electronic interactions?

Edit: As pointed out by Sandejo in the comments the assumption that the electron is at a particular position is wrong, however even if we leave out that part i still feel the symmetry argument should hold

FoundABetterName
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No. You seem to be mixing up the concepts of average position and the center of mass. It is not as if the electron 'fills up' space all around the center of the atom. If that were the case, one perhaps could make an argument that the 'center of mass' of this is at the nucleus. However, this is emphatically NOT the case: the electron is a point particle with ill-defined position and momentum. When one tries to measure the position, one can say the center of mass is wherever you measured it, but outside of this it does not make sense to ask where the center of mass is; after all, the position itself is not defined then.

John Dumancic
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  • But as a comment pointed out and as far as I know it acts as a wave as well as a particle right so is there any sense of a centre of mass in a wave context? – FoundABetterName Aug 03 '20 at 18:14
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    You can certainly define the center of mass of a wavefunction in this way. However, this is not the usual definition. In physics, "center of mass" is ordinarily used only in a classical context. "Expected value of position" is the usual term for the concept you're describing. Perhaps you could edit your question to make the language more standard? – Daniel Aug 05 '20 at 19:27
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    However, this only rescues parts 1-4 of your question. Parts 5-7 are confused about more than just definitions. Fortunately, 1-4 are the interesting parts. – Daniel Aug 05 '20 at 19:29
  • @Daniel I don't have much experience with the wave functions and QM in terms of solving them So I don't really know what changes to make and can you link some sources on how to find the COM of a wave function as you mention – FoundABetterName Aug 06 '20 at 00:53
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    Like I said, what you call "center of mass" is more commonly called "expected value of position." See the first two paragraphs of http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html for a very brief explanation of expected values, or almost any intro QM textbook for a full explanation. – Daniel Aug 06 '20 at 02:43