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The time average position of a 1s electron is most of the time out of the nucleus because even though the maximum of the wavefunction is at 0, <psi|r|psi>=0 for r=0. But as the nucleus isn’t point like, there is still a probability to find the electron inside the nucleus. <psi|r|psi>=/=0 for r close to 0 but not 0.
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