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Why can electrons not exist between shells?

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They can. It takes a finite time for an electron to make a transition. They just can't stay between the shells, because there is no energy eigenstate there, and only energy eigenstates are stationary. read more

Its not true. Electrons can exist between shells. When electrons are bound to a nucleus, it is true that the only stable states are the ones with discrete quantum numbers - corresponding to the "shells" you mention. read more

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