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Yes, as the title states: Do orbitals overlap ?

I mean, if I take a look at this figure...

enter image description here

I see the distribution in different orbitals. So if for example I take the S orbitals, they are all just a sphere. So wont the 2S orbital overlap with the 1S overlap, making the electrons in each orbital "meet" at some point? Or have I misunderstood something?

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    $\begingroup$ If you take 2 orthogonal states $\phi$ and $\psi$, $\langle \phi|\psi \rangle = 0$ means $\int dx \bar \phi(x) \psi(x) =0$, but it does not necessarily means that the domains where $\phi$ and $\psi$ are not null, are different space domains (with no overlap). $\endgroup$ – Trimok Oct 8 '13 at 9:34
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If you mean to ask "do the orbital radial probability distributions overlap?", the answer is yes:

enter image description here

Image Credit

making the electrons in each orbital "meet" at some point

As you can see from the image, the electron orbitals are not position eigentstates. If you're imagining two point-like electrons in different orbitals colliding, you're not thinking "quantum mechanically".

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  • $\begingroup$ But your image shows the probability, right ? So isn't there, even though the probability are getting smaller and smaller for higher $n$ near the nucleus, that they will actually "meet" at some time ? I know that it is probabilities of where the electrons can be, but if I make a gazillion measurements, at some point and time, wouldn't I be able to measure two electrons, of two different orbitals, being in the same place (Although I guess Heisenberg Uncertainty Principle has to be fulfilled) ? $\endgroup$ – Denver Dang Oct 8 '13 at 0:35
  • $\begingroup$ @DenverDang, you're still not thinking in QM terms. An electron in an orbital does not have a definite position. Moreover, two electrons cannot be in the same state (Pauli Exclusion Principle) so, no, you could not measure two electrons to be in the same place for that would mean that two electrons have the same state. $\endgroup$ – Alfred Centauri Oct 8 '13 at 0:39
  • $\begingroup$ It is worth noting that the displayed orbitals are for the solution to hydrogen-like atoms, which means that they are only correct if there is only one electron. Once there are more than one electron the situation becomes much more complicated, thought there do exist exact numeric solutions for various states of several-electron atoms. $\endgroup$ – dmckee --- ex-moderator kitten Oct 8 '13 at 1:20
  • $\begingroup$ @dmckee, a worthy note indeed. $\endgroup$ – Alfred Centauri Oct 8 '13 at 1:25
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    $\begingroup$ @AlfredCentauri, I guess you mean that observing the electron at a definite location collapses its wave function to a Dirac delta, so that if two electrons are observed at the same location, both of their wave functions would collapse to the same delta function. I think indeed this cannot happen for several physical reasons (like the inter-electronic potential), but I don't think the Pauli exclusion principle is one of them, as spin is part of the state, so that electrons with different spin can have identical position wave functions. Please correct me if I'm wrong! $\endgroup$ – doetoe Oct 8 '13 at 8:35
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An orbital is essentially a wave function from which a probability distribution of the location of an electron upon measurement can be inferred. What is depicted will be something like the region within which the probability is 50% (shaped in a way that depends on a decomposition of the state function in a radial part and an angular part).

If you mean to ask if these regions overlap, yes, they certainly do. If you mean to ask if the regions of space where the probabilities are non-zero overlap, they even more certainly do, as the probability is non-zero almost everywhere (i.e. zero on set of volume 0). If you mean to ask if they (or rather the spaces they span) in the state space overlap, then no: see Programming Enthusiast's answer.

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The orbitals shown in the figure are the different eigenstates of the electronic wave functions derived from the solution of Schrodinger equation. One of the postulates of QM states that these eigenstates are independent in a free atom.

Orbitals do overlap when two atoms are close together, e.g. in a molcule and the degree of overlap corresponds to the type of bonding (ionic, covalent etc.).

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Yes, orbitals do overlap. However, these orbitals do not necessarily mean that is where the electron is. The orbitals are only the probable locations that an electron would be. In fact, electrons are not even necessarily in those orbitals; they could be anywhere. This is able to be shown mathematically, but it is relatively dense stuff. If you are curious and want a detailed explanation, I would be happy to edit the answer to show all of that.

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There has been a lot of emphasis on QM when detailing events here. Firstly, it would be beneficial to examine the foundations to modern physics, which are solidly hinged on mathematical pretexts that show significance when relating to various physical phenomenon. Quantum mechanics dwells on mathematical significance as perspectives. But that is not to say, that physical aspects to such measurements can be discounted, or misrepresented in a qualitative analysis.

The waves representative to particles mentioned herein represent radial probabilities, that are cumulative in a sense; rather than measuring electron densities at various points, the objective is to analyse the radial probability which takes into account the cumulative probability of an infinitesimally small volume at a distance r from the center of the nucleus, extended 3 dimensionally over a region encapsulating the nucleus at that distance. This eliminates the need to define the precise location of the electron, and thus, extends the uncertainty to a larger region by focusing on the probability distribution in the region. Hence the observation of the uncertainty principle.

All of the above remains valid so far as measurements and analyses are concerned. Ultimately, there is that certain electron in n=1 and n=2 and so on, and based on the probability distributions, these particles can exist at the same place, but probably at different times, if one is to observe other principles associated to the state of the electron.

Thats right; electrons do have definitive positions at a given time. The uncertainty principle governs measurements only, and is a correct one; we cant measure the position and momentum of an electron precisely, all at once.

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