# Field lines of electrons in an atomic orbital

Hello,
I have a rather conceptual question I couldn't really find an answer to yet.
The electric field lines of an isolated resting electron would simply point from everywhere in space towards the position the electron is located in, with a length inversely proportional to the squared distance to that electron position.
What I'm wondering now is how electric field lines would look like for an electron in an hydrogen atom. So if one would subtract the influence of the positive proton on the actual hydrogen field lines, would the (electric) field lines of an electron in an 1s orbital still have that same shape as an isolated electron (resting at the position of the nucleus) would have ? How about the 2s orbital ? Does the length of the field lines go to zero at the radial node ? How about p-orbitals ? Where do the field lines even point to in this case ? Or is the electric field simply the integral of the isolated electron field lines and the electron density distribution of the orbital (so the field lines do not go to zero at the nodes of the orbital) ? I'm struggling with picturing this conceptually. Or is my question ill-posed ? Maybe someone could help me out here ?

mfb
Mentor
It is problematic to mix classical concepts (field lines) with quantum mechanics (electron orbitals). For some setups, you can treat the orbital as a charge distribution and calculate field lines just based on that. For distances large compared to the orbital you get the same (independent of the type), close to the orbitals you get something different. Close to the nucleus, the field (from the electron alone) would drop to zero, indeed.

Well, but isn't the electromagnetic field a set of 2 3d-vector-fields (the E and B fields) ? so just considering the E field the vectors must point in some direction and have some magnitude. That's what I mean by "electric field lines" - maybe that was confusing. So maybe it makes more sense to say: How does the E-field look like for those orbitals considering the electron alone ?

It is problematic to mix classical concepts (field lines) with quantum mechanics (electron orbitals). For some setups, you can treat the orbital as a charge distribution and calculate field lines just based on that. For distances large compared to the orbital you get the same (independent of the type), close to the orbitals you get something different. Close to the nucleus, the field (from the electron alone) would drop to zero, indeed.

so there is no way to compute a electro-magnetic (near) field from those orbitals ?

sophiecentaur
Gold Member
so there is no way to compute a electro-magnetic (near) field from those orbitals ?

That would be difficult and not a valid concept when you think that the electron is not actually 'anywhere' in particular when it's in a bound state. It behaves like a standing wave in many respects so how could you assign it a position and direction in order to treat it the same as an electron in free space, as in a CRT?
You have to break free from the concrete thinking of classical Physics.

PS They are called "orbitals" and not 'orbits', for a good reason. Electrons are not in orbit.

That would be difficult and not a valid concept when you think that the electron is not actually 'anywhere' in particular when it's in a bound state. It behaves like a standing wave in many respects so how could you assign it a position and direction in order to treat it the same as an electron in free space, as in a CRT?
You have to break free from the concrete thinking of classical Physics.

PS They are called "orbitals" and not 'orbits', for a good reason. Electrons are not in orbit.

I'm aware that they're spread out over the whole orbital and not just an orbiting point, but i thought it should still be possible to compute a concrete electrical field. Maybe it could apply that the resulting electrical field would be sort of the convolution of the electron density function of the orbital with the electric field of a stationary electron (the inverse square law). So in effect it would be like chopping up the electron's charge into infinitesimal parts, distribute them over the orbital accourding to its electron density function and compute the electrical field for each of these (infinite) parts of charge and accumulate over all of them.
That was just an idea, though. Is this so far off conceptually ? Sounds fairly reasonable, I think. :)

@mfb: after re-reading your post, it sounds like you suggest something similar (i think i misunderstood at first - so don't mind my first reply to your post ;) but maybe you can approve/disapprove if the way I explained in this post would yield what you meant by "distributing the charge"), but can you clarify conceptually for which setups it would work and in which cases it'll break down ?

Thanks for all the input so far :)

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sophiecentaur
Gold Member
I'm aware that they're spread out over the whole orbital and not just an orbiting point, but i thought it should still be possible to compute a concrete electrical field. Maybe it could apply that the resulting electrical field would be sort of the convolution of the electron density function of the orbital with the electric field of a stationary electron (the inverse square law). So in effect it would be like chopping up the electron's charge into infinitesimal parts, distribute them over the orbital accourding to its electron density function and compute the electrical field for each of these (infinite) parts of charge and accumulate over all of them.
That was just an idea, though. Is this so far off conceptually ? Sounds fairly reasonable, I think. :)

@mfb: after re-reading your post, it sounds like you suggest something similar (i think i misunderstood at first - so don't mind my first reply to your post ;) but maybe you can approve/disapprove if the way I explained in this post would yield what you meant by "distributing the charge"), but can you clarify conceptually for which setups it would work and in which cases it'll break down ?

Thanks for all the input so far :)

But you would need to do a vectorial addition of all those constituent fields. This would have to assume that they were all in every spot at the same time? (Are they?) I can't see how any calculation you might do would be likely to yield a meaningful result. You seem to be after a classical description of something that is just not classical. Why not make the leap and accept that QM rules in these circumstances? And, whilst you are about it, what about the field due to the Nucleus? The electrons are only doing what they do in atoms because of the presence of the other charges in the atom.

mfb
Mentor
The concept of the quantum defects treats all electrons (apart from the one where the energy is calculated) as effective contribution to the potential the electron sees, and as far as I know the Hartree-Fock method uses a similar approach for each electron.

xortdsc said:
it sounds like you suggest something similar ([...] but maybe you can approve/disapprove if the way I explained in this post would yield what you meant by "distributing the charge")

but can you clarify conceptually for which setups it would work and in which cases it'll break down ?
I don't know, but maybe the two links above can help.

But you would need to do a vectorial addition of all those constituent fields. This would have to assume that they were all in every spot at the same time? (Are they?) I can't see how any calculation you might do would be likely to yield a meaningful result.

Well, yes there is a vectorial addition implicitly. I'd mean to convolve the electric field of a single free/isolated electron (3-vector field) with the normalized electron density function of the orbital (scalar field whose integral is unity so total charge will be conserved) which would yield a new convolved 3-vector field which is ought to be the electric field of the electron in the orbital.
And yes, it would assume the electron is everywhere in that orbital at the same time, which I think is how it is supposed to be interpreted (no "moving electron" but a "stretched out electron" instead). At least that what I keep reading everywhere :)

The concept of the quantum defects treats all electrons (apart from the one where the energy is calculated) as effective contribution to the potential the electron sees, and as far as I know the Hartree-Fock method uses a similar approach for each electron.
I don't know, but maybe the two links above can help.

This seems to consider multi-electron atoms which are much more complicated. I'm just talking about a single electron hydrogen atom.

mfb
Mentor
I'm just talking about a single electron hydrogen atom.
You are talking about the electric field of a single electron. Where is the point in knowing that, if you don't want to consider its effect on other particles? Other electrons in the atom are those particles...

sophiecentaur
Gold Member
xortdsc
The only time that the field around an electron is calculable (or meaningful) is when it is isolated. In the presence of just a single proton (bound in a Hydrogen atom) it (if you wanted to consider it) will be severely modified and asymmetrical.
In a similar, very classical vein, you must have seen the magnetic field of a bar magnet and then seen how it changes when another magnet is brought close to it. In that situation it's the field of both magnets that is seen and not just that of the original magnet.
Why not just accept that it's an invalid notion with little or no relevance and not any more knowable than the actual 'position' of the electron at any time.

I'm sorry to insist that something sounds wrong to me when you say it is impossible or wouldn't make any sense.
Mainly because the fact that the electric field of multiple (isolated) electrons can be computed easily by super-positioning their individual fields - so I don't really see how you can say that you can compute the field for a single magnet, but not for two - at least for charges this works nicely and I think so it should for magnets.
And for another example which is closer to my question: Consider molecular dynamics simulations; They are very accurate in the resulting dynamics so there must be some way to compute the charge distribution (and field) of those molecules to compute their interaction. How could this be possible if even for the simplest possible case, a hydrogen atom in the ground state, it would be impossible ? They seem to be able to somehow solve a much more complicated problem where I'm only interested in a single hydrogen atom.

sophiecentaur
Gold Member
There is just one electron BUT where is it? You could say it is in random places around the nucleus. How does that help you to achieve anything? In fact what is it all supposed to mean?

regardless of whether the electron pops up in random places very quickly or is in a stable distributed state, the resulting time-averaged electrical field it produces should still be identical and stable, i'd think. it's this (averaged) field i'd be interested in (as it gives rise to interactions).

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sophiecentaur
Gold Member
regardless of whether the electron pops up in random places very quickly or is in a stable distributed state, the resulting time-averaged electrical field it produces should still be identical and stable, i'd think. it's this (averaged) field i'd be interested in (as it gives rise to interactions).
Are you suggesting that the various 'bits' of the electron are located around the atom in a fixed pattern? Surely the whole point of QM is that such things are totally random in time and space.

I really can't see how such a random set of vectors, from each of the constituent bits, each one varying in time (according to a random presence or absence of a charge in any one particular pace), could combine to produce a stable field. I think it's accepted that there is a sort of fuzzy statistical distribution of Potential around an atom or molecule - which is what the polarity of molecules is based on. Field, otoh, is a different matter.
But this model is really far too classical to be workable. You start with QM, to get your charge distribution (is that even a valid step?) and then revert back into classical for working out a field. It doesn't seem valid to me.

no, i'm not assuming a fixed pattern. only a distribution (which is sort of the averaged presence of the electron for each point in space). so the basis of the calculation should be this averaged presence, disregarding the random nature of the actual process.

sophiecentaur
Gold Member
no, i'm not assuming a fixed pattern. only a distribution (which is sort of the averaged presence of the electron for each point in space). so the basis of the calculation should be this averaged presence, disregarding the random nature of the actual process.

There's your problem. If you disregard the random nature then your resulting numbers would have no connection with reality. And what magnitude and direction would you give the resultant of a random set of vectors? If you don't take the direction and amplitude of each vector into account then you will get nonsense. If you really believe in this, I suggest you sit down and try to draw out a simple version of what you would actually do. Start with a simple model of two charge centres with a range of different amplitudes, varying in time. Add the vectors at different times and see what you get. Then do the same with three charges - and so on. The result will tend to the field of a single charge (the sum) at a distance (i.e. the same as an isolated electron). But don't you really need the Proton charge there, too? The field, then would go to zero at infinity.
I still think the Potential distribution would be a more meaningful thing to go for. For a start, it would be easier / possible to plot.

mfb
Mentor
Quantum mechanics is not random, it is a deterministic theory. The interpretations of measurements can be, but if we consider processes where those measurements do not happen everything is certainly deterministic.

sophiecentaur
Gold Member
Quantum mechanics is not random, it is a deterministic theory. The interpretations of measurements can be, but if we consider processes where those measurements do not happen everything is certainly deterministic.

OK So where does this take the problem of adding vectors? Does it make the idea more valid - or less?
My brain hurts with QM but it hurts even more in trying to apply classical ideas in the context of QM.

Dale
Mentor
2021 Award
Hmm, I don't know about the idea that the field is undefined or anything. The field is a measurable quantity, so there should be some operator that you can apply to the wave function to get the distribution for any measurement of the field. Of course, I would assume that the wavefunction isn't usually in an eigenstate of the field, so the best you could do would be to get an expected value.

mfb
Mentor
OK So where does this take the problem of adding vectors? Does it make the idea more valid - or less?
My brain hurts with QM but it hurts even more in trying to apply classical ideas in the context of QM.
Where is the problem? For each point in phase-space, you calculate the contribution to the fields, and then integrate over the whole phase-space, with the squared amplitude as weight.

sophiecentaur
Gold Member
Where is the problem? For each point in phase-space, you calculate the contribution to the fields, and then integrate over the whole phase-space, with the squared amplitude as weight.

Does that not require you to be sure that the charge is actually in existence all the time? This is not how I understood the probability density of the bound electron; I thought there was a random element. What is the modern opinion on this?

mfb
Mentor
The electron and its charge exist all the time.

Hmm, I don't know about the idea that the field is undefined or anything. The field is a measurable quantity, so there should be some operator that you can apply to the wave function to get the distribution for any measurement of the field. Of course, I would assume that the wavefunction isn't usually in an eigenstate of the field, so the best you could do would be to get an expected value.

That was exactly my way of reasoning. It is measurable so it should be computable, except the theory is incomplete on a fundamental level. :)
The expected (idealized) value would be good enough for me.

Where is the problem? For each point in phase-space, you calculate the contribution to the fields, and then integrate over the whole phase-space, with the squared amplitude as weight.

That's what I thought, but after further thinking about it there really IS a problem (merely a computational one): For a traditional point-like charge the field goes to infinity at the position of the charge, independent of its magnitude (which is possibly not 100% correct, but all we have in the classical theory). So, if one would use the charge density as a source and apply that traditional law to the partial charges it would yield infinity everywhere. So for this reason it may really be impossible. :/

But isn't there a better way of doing it ? Is QM really that incomplete when it comes to the EM-fields it gives rise to ?

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