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Let's have two electrons in two different atoms separated by a distance. For instance, let's consider one atom in New York and another atom in San Francisco. Can they have the same quantum number? Can they have the same energy level?

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    $\begingroup$ Yes. They are not in the same state because their wavefunctions will be centered on different coordinates (the coordinates of their respective nuclei). $\endgroup$ Nov 29, 2023 at 11:38

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The answer is "yes, but". The state can be $$ \left(|\phi,NewYork\rangle_1 |\phi,SanFran\rangle_2 - |\phi,NewYork\rangle_2 |\phi,SanFran\rangle_1 \right) / \sqrt{2} $$ where $\phi$ is the internal state and the subscript 1,2 labels the electrons. So you can say there are two electrons here, and they have the same internal state. But you can't say which electron has which location unless you refer to them by location. So you might say "an electron in New York was observed," or evolved in some way, or whatever. But you can't say "and that was electron number 1."

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Strictly they cannot. Two such atoms would produce a bonding and an anti-bonding orbital, leading to singlet $$ \left( |a \overline{b} - b \overline{a} \gt \right) / \sqrt{2} $$ and triplet $$ | ab \gt \,.$$ For all practical purposes, of course, $a$ and $b$ are degenerate at any distance such that single atomic orbitals do not overlap appreciably and the overlap integral vanishes very fast with distance beyond a few ångström.

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