I have read multiple answers on StackExchange about this question, but I wasn't able to find a concrete answer. Like other questions, the reason I ask about the $s$-orbital is because it has a zero orbital angular momentum. But, the implication of having a zero angular momentum is unclear? Some answers discuss probability distributions, but the question is how can there be multiple places for an electron, if it can't move. I read another answer that says that the electron passes through the nucleus or curve around it as a wave. I would appreciate it if anyone could provide a resolution to this question.
Answer
For me the physical implication of zero angular momentum is that the electron's probability distribution is spherically symmetric. At the deepest level, the angular momentum property in quantum mechanics describes how something transforms under rotations (see Noether's theorem). Although this is quite an abstract property, in the case of electron orbitals it relates to something extremely concrete and amenable to visualisation: namely, the shape of the angular probability distribution.
Any interpretation based on classical concepts is doomed to fail at some point. In this case the confusion arises from imagining that the electron is in a certain "place", or that it "moves". The only truly useful description of the electron that has been found is the quantum wave function, and its corresponding probability distribution.
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