How Do Electrons Behave in a Half-Filled P Orbital?

[DiracPool]

I recently have been studying how the SE solutions produce orbitals in the atom. I understand that we are not to think of these classically as actual orbits, however, I am looking for a little clarification on a couple things. For one, it is relatively easy to see how two electrons may share a 1s spherical orbital. But what about a p orbital? Is each electron confined to one or the other lobes of the dumbbell? After all, there is a zero probability that the electron can be found between the lobes and therefore a zero probability that it can smoothly transition (classically at least) between the two.

I know that this is QM and that we are supposed to allow for these counterintuitive things to happen, but has there actually been any experiments that conclusively demonstrate that, in a half-filled p orbital, the SAME electron is found zipping around both lobes of the orbital. Or is it typically found in just one? In addition, are there any experiments that tell us whether two electrons filling a p orbital can be found in either lobe? Or that both can be found in one lobe at the same time and not the other? I’m guessing that if measurements have found that both electrons demonstrating opposite spins have an equal probability of being in either lobe at any given time, then I’ll have to accept that. I’m just looking for a less counterintuitive solution or visualization.

 

[Vanadium 50]

First, there is no way to tell if two electrons are the same or different, other than measuring both positions at the same time. Most measurements look at a time-averaged charge density.

One can calculate the position of one electron in the frame where the other electron is at 0,0,0. I expect it's a numerical mess, and I suspect that most of the time it is in the other lobe, but not always.

 

[Sonderval]

"After all, there is a zero probability that the electron can be found between the lobes and therefore a zero probability that it can smoothly transition (classically at least) between the two."
Be careful. This is a probability density, not a probability. The prob. density vanishes only in a set of measure zero, so your intuitive picture is not really mathematically well-defined. I think.
If you have two electrons in the same p-Orbital, each has the same probability density (ignoring the complication of Coulomb repulsion between the electrons).
BTw, it's the same with the higher s-Orbitals - they all have radius values where psi vanishes.

 

[Bill_K]

Is each electron confined to one or the other lobes of the dumbbell? After all, there is a zero probability that the electron can be found between the lobes and therefore a zero probability that it can smoothly transition (classically at least) between the two.

No, as you say the orbital represents the probability of finding the electron at a particular point. The fact that it vanishes at z = 0 simply means that you won't ever find it at z = 0. But you can find it on either side - the electron does not have a continuous orbital motion, and therefore does not have to transition from one lobe to the other. It can be found in either lobe.

has there actually been any experiments that conclusively demonstrate that, in a half-filled p orbital, the SAME electron is found zipping around both lobes of the orbital. Or is it typically found in just one?

It don't zip, it just sits there! No, there is no doubt about this. The orbital is symmetrical in the z direction. If the electron occupied just one of the two lobes (choosing a lobe at random?) there would be two possible states, not one, which would be in total conflict with observation.

 

[DiracPool]

It don't zip, it just sits there! No, there is no doubt about this. The orbital is symmetrical in the z direction. If the electron occupied just one of the two lobes (choosing a lobe at random?) there would be two possible states, not one, which would be in total conflict with observation.

So, would it be accurate to say that a single electron in a p-orbital is not zipping around the orbital? It just simply occupies both lobes with a charge density distributed according to the probability density? And then when a second electron comes around the first electron simply accommodates it through both electrons mutually orienting their "spins" orthogonally to one another (however that trick may be manifested)?

In that scenario, then, I would assume that both electrons occupy both lobes of the orbital with a relatively equal charge density distribution distinguished only by some spin factor? Again, in that scenario, is there any way to distinguish one electron from the other, i.e., is there some inhomogeneity of the charge density distributions that may distinguish one electron from the other? Or, is there no way to distinguish them and we simply say that two electrons share this orbital with opposite spins and that's about all we can say about it?

 

[Sonderval]

"that's about all we can say about it?"
Yup