According to Pauli’s exclusion principle, an $s$ orbital contains at most two electrons with the opposite spin (up and down). Why can't an $s$ orbital contain a third electron whose state is the linear combination of spin up and down?
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Because we can have only one electron per quantum state. Spin up and spin down are two different states. A linear combination of the two is not a new independent state. It is obviously formed from the spin up and spin down states. |
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