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Can an electron occupy an energy level lower than its ground state? Do electrons come closer to each other at 0K temperature?

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    $\begingroup$ What do you mean by ground state other than 'lowest energy level'? Obviously you can't have a lower energy than the lowest allowed energy. $\endgroup$ – jacob1729 Jun 1 at 16:37
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    $\begingroup$ Yes, this is all the answer this question needs. It's not some physics principle but just understanding the meaning of the words used. $\endgroup$ – nasu Jun 2 at 13:15
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No quantum particles can't posses an energy lower than the ground state of the system, see variational method for some more detail on this. https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)

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The other answer is correct, no electron can occupy as per QM an energy level lower then the ground state. I would like to add a few things.

It is very important to differentiate between classical orbit and QM orbital. The classical (planet like) orbit for electrons is no longer valid, now we use QM orbitals, that state that the electron exists around the nucleus at a certain energy level.

There are three forces that govern the electrons at the energy level:

  1. the EM force keeps the electron close to the nucleus (neutrons)

  2. the kinetic energy of the electron keeps it away from the nucleus

  3. the Heisenberg uncertainty principle keeps the electron away from the nucleus

When these three forces equal out, the electron is at a stable energy level as per QM.

The reason why it is so, is because:

  1. as soon as the electron would try to come closer to the nucleus, that is to be restricted to a space volume smaller then the ground state, that is a QM orbital closer to the nucleus then the ground state, its position would be known with higher certainty, and the Heisenberg uncertainty principle would cause the electron to gain momentum (its momentum would be known with lesser precision), thus the electron would move away from the nucleus
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