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In the following figure enter image description here

we can see that the p-orbitals overlap 1s orbital (though relatively very little). How can an electron in p-orbital, be simultaneously in the 1s orbital at any given point in space and time?

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The question is interesting, but slightly ill-defined. If you take s and p orbitals as a solution of Schrodinger equation of hydrogen-like atom, then they are eigenfunctions of Hamiltonian, therefore are orthogonal. That means - no overlap. Do not be misled by the figure, if you do the math properly, $\langle1s|2p\rangle = 0$.

If you speak of just one electron, the solution is reasonably simple as stated in the article and the electron does not simultaneously reside in both orbitals. Even if you would think of interaction with excited states, it would not help, as single excitations do not interact with ground state. You can of course excite it, but than it is not "simultaneous".

In many-electron system, the picture gets more complicated, but this was not a question.

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It is important to know that the overlap must be performed with the wave functions $\psi_j(\vec{r})$, not the density, $\rho_j(\vec{r}) = \psi_j(\vec{r})^* \psi_j(\vec{r})$. Wave functions have signs; they oscillate, as these oscillations encode the momentum distribution $\rho_j(\vec{p})$.

When you calculate the overlap between $\psi_s(\vec{r})$ and $\psi_p(\vec{r})$, $\int_V \psi_s(\vec{r}) \psi_p(\vec{r}) dV$ ($V$ is the volume) you should remember that the lobes of the "p" wave function have different signs, while the spherical $s$ wave function keeps the same sign over the surface of the sphere. Therefore, the integral has two identical but opposite (different sign) contributions that cancel the overlap.

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Remember that the orbitals are really just approximations showing the most likely places for an electron with a given set of quantum numbers to be. If you don't apply a cutoff (like less than a 1/1000000 chance), all orbitals would encompass the entire universe. We just make pictures like this by identifying the places that an electron could be within some margin of probability.

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