# Classical spin of an electron?

Listening to Susskind's online QM lectures, he has mentioned the "classical spin" of an electron a few times (with an associated magnetic moment), but I didn't think the electron really has a classical spin.

Specifically, in Maxwell's equations we only get magnetic fields for moving charges, so I don't see how a point particle like an electron would have a magnetic moment in a classical context if it is at a fixed position in space.

The only way I can think of a electron with spin in a classical context would be if you modelled it as some specific charge distribution in a volume and then set that charge distribution spinning. Then add up the magnetic field contributions from all the individual bits of that moving distribution, to achieve a classically model of a spinning electron with Maxwell's equations. But if you did this it would radiate (like a classical electron in the Bohr model should), and I presume would eventually loose its angular momentum and spin to that radiation.

Can anybody guess what Susskind may be talking about here?

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Actually In the early twenties before Pauli, there were models of the electron being a spinning particle, and these were made by well known physicists, like Kronig (the same guy from the Kramers-Kronig relation), but after doing calculations, In this model the "edge" of the electron should have had greater speed than that of light and there were many other discrepencies.
So later building on the stern-gerlach experiment, Pauli built his theory of this so called "spin" which was as I wrote above considered earlier as the "true" angular momentum of the electron itself. In this he introduced the famous Pauli-matrices. And concluded that this was a purely quantum mechanical effect. But this was a phenomenological description, which even Pauli himself dubbed mysterious.
But 2-3 years later, Dirac formulated his equation for the relativistic electron, and from there the spin arose naturally as a quantum mechanical effect.

I dont believe that the electron really spins but why would it radiate? the field isnt changing.

I was imagining a sphere of spinning charge. None of that is moving linearly, so there would be radiation associated with the acceleration.

I should go back and relisten to the lectures in question and note down exactly what Susskind said.

There are two critical problems in the electron spinning model.

The size of the electron is too small. So by equating the angular momentum of the spinning sphere of the electron to 1/2 h-bar, the sphere speed leads to about one hundred times the speed of light.

And the spin g factor is about 2. This means the unequally distribution of the charge and mass of the electron.
(If the charge and mass are equally distributed, the g-factor is usually 1. So the spin g-gactor 2 is remarkably different.)

Considering the electron size, If we suppose that the whole electron cloud is spinning,
we can not explain about the spin g-factor 2.
( The mass and charge of an electron are separated !?).

separate charge and mass could conceivably explain why the electron (seems) to fill atomic orbitals out of order (filling higher orbitals before lower ones)

There are two critical problems in the electron spinning model.

The size of the electron is too small. So by equating the angular momentum of the spinning sphere of the electron to 1/2 h-bar, the sphere speed leads to about one hundred times the speed of light.
So what is this electron size you talk of. Are you referring to the classical electron radius? This size is based on the assumption that the electron mass is entirely due to it's electrostatic potential energy. This assumption is not necessarily true, therefore you can not rely on this as a solid basis to eliminate the spinning electron model.

As grandpa has mentioned repeatedly, although an accelerated point particle radiates, an extended distribution of charge can accelerate without radiating.

So what is this electron size you talk of. Are you referring to the classical electron radius? This size is based on the assumption that the electron mass is entirely due to it's electrostatic potential energy. This assumption is not necessarily true, therefore you can not rely on this as a solid basis to eliminate the spinning electron model.

As grandpa has mentioned repeatedly, although an accelerated point particle radiates, an extended distribution of charge can accelerate without radiating.
Ok. Then how do you explain about the spin g-factor 2 ?
The mass and charge of one electron are separated !?

And if the whole electron cloud is spinning, what state is the center of the cloud ?
The orbital angular momentum in the ground state in the hydrogen atom is zero.
Why is not the electron scatterd by the nucleus.

Matterwave
Gold Member
So what is this electron size you talk of. Are you referring to the classical electron radius? This size is based on the assumption that the electron mass is entirely due to it's electrostatic potential energy. This assumption is not necessarily true, therefore you can not rely on this as a solid basis to eliminate the spinning electron model.

As grandpa has mentioned repeatedly, although an accelerated point particle radiates, an extended distribution of charge can accelerate without radiating.
The actual radius of the electron is actually smaller than the radius you speak of, which only makes things worse...

An electron HAS an actual radius?
This is news to me...

Matterwave
Gold Member
I meant the upper bound to the actual radius, sorry, slip of the tongue.

We can not measure the correct size of an electron.
Because an electron is too small to find a proper smaller particle in the scattering experiment.

If one electron is as big as the electron cloud, we can measure the size of an electron using a proton or something in the scattering experiment.

Matterwave
Gold Member
Yea, what an electron ACTUALLY is is unknown. Is it a 1-D loop of string vibrating at some mode? Or is it a point? Or some other? Who knows. In many of these cases, a "radius" is not even well defined. The point I was making is that IF you view the electron as a really small ball that's spinning, the upper bound to that radius is smaller than the classical electron radius (more pedantically, the mass/energy of the electron is confined to a smaller region of space).

My semantics may have been wrong, or overly misleading, and I apologize for that, but the point I was making still stands. But do forgive me because it would take forever (and be rather annoying) if I typed every post in a completely pedantic sort of way...

most of the experiments 'measuring' the size of the electron just measure g (gyromagnetic ratio). its agreement with the calculated value is taken as proof that the electron is a point particle.

if the electron is a spinning sphere of charge then it is quite trivial to determine what radius it must have. we know that the frequency at which an electron in the ground state of hydrogen rotates is R (Rydberg constant). we know the magnetic moment and so from the definition of magnetic moment the required radius can easily be determined. (not forgeting that g is not equal to 1)

the equation for determining magnetic moment is the same as for determining angular momentum. just replace mass with charge.

is the idea that the mass and charge occupy different orbitals any more strange than any other quantum phenomena (like superposition or entanglement)?

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f95toli
Gold Member
Can anybody guess what Susskind may be talking about here?
Maybe he is just refering to the fact that much of the physics can be modelled as if the electron had a "classical" spin with angular momentum $\hbar L$.

If you look in a book about e.g. NMR you will find a LOT of classical physics, this does not mean that anyone believes that this is an accurate description of what is really going on(for that you need QM), but in many cases we can "pretend" that we are dealing with classical systems since the formalism is much simpler; it also tends to be quite accurate if we are for example dealing with large ensembles at elevated temperatures (where the "quantumness" of the individual systems tends to be washed out anyway).

Granpa: I think there are actually high energy scattering experiments which place an upper bound on the actual physical size of the electron as 10 ^-18 m. What do you mean by "mass and charge occupuying separate orbitals"? And when does an electron fill higher-energy orbitals in preference to lower ones?

I thought so too until I started looking into it. the experiments I read about arent as impressive as they are made to sound. I have become rather skeptical of experiments that people toss out as 'proof' of some idea without ever bothering to explain exactly how the experiment works and exactly what the raw data from the experiment was.

separate mass and charge is explained elsewhere in this thread. no point in me repeating myself endlessly.

higher as in further from the nucleus. that should have been obvious.

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Can anybody guess what Susskind may be talking about here?
Well you could write down the dirac lagrangian and not quantize it. The field would then be a classical spinor field. In this sense the electron field could still be said to have spin. This would also correspond to doing a calculation at tree level so with no quantum corrections. this i think gives g=2. The loop corrections then add the decimal corrections to the 2. Both classically and after quantization spin is still defined at a point. An electron is a point particle so it doesn't have a size.

I thought so too until I started looking into it. the experiments I read about arent as impressive as they are made to sound. I have become rather skeptical of experiments that people toss out as 'proof' of some idea without ever bothering to explain exactly how the experiment works and exactly what the raw data from the experiment was.

separate mass and charge is explained elsewhere in this thread. no point in me repeating myself endlessly.

higher as in further from the nucleus. that should have been obvious.
As it stands the statement "mass and charge occupying separate orbitals" doesn't seem to make much sense to me. As properties of the electron, they're parameters that partly determine the solution to the Schroedinger equation, from which one can extract the probabilistic distribution of the position of the electron, i.e. the orbitals. They don't have a position, or a momentum, or any similar dynamical variable. Only the electron itself does. How would you describe such an idea mathematically?

Whilst I understood what you meant by a higher orbital, I've never heard of a higher orbital being filled in preference to a lower one. Anywhere I could read an account of this being measured? And does the system decay to the ground state in fairly short order?

valence electrons are always in the highest shell (and are always s and p orbitals). transition elements are elements that go back and fill in d orbitals of lower shells.

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Oh, of course... It's been a while since A-level chemistry, sorry
Why do you think that endowing mass and charge with their own orbitals (whatever that would mean) would explain this? Doesn't it already have an explanation in terms of the central field approximation? I just dug out my 2nd year atomic physics lecture notes- it's asserted that one exists, although it isn't presented to us.
Also, do you have any references I could read about the size of the electron being inferred from scattering experiments?

the whole point of the idea of valence electrons is that only they interact in chemical reactions because all other electrons are within their orbitals and are therefore shielded.

the idea isnt complicated. I imagine that it has occurred to most people. if the electron doesnt always go into the lowest orbital then maybe the electron consists of 2 parts one of which does always go into the lowest orbital while the other doesnt.

in this model the frequency of emitted light would be the difference of the frequencies (rotation rates) of the 2 orbitals that the electron transitions between therefore the 'energy' of the orbital would correspond to its rotation rate which would be determined entirely by the size of the mass orbital (due to conservation of angular momentum). specifically, frequency=1/r^2 where r is any integer. since electrons fill orbitals in order of decreasing energy it follows that the mass (unlike the charge) would always go to the lowest available orbital.

http://en.wikipedia.org/wiki/Onium
(I havent thought this part through entirely so feel free to contribute)(I'll no doubt get ripped to shreds here).
since their charges cancel out, such particles imply that the mass also experiences a force pulling it toward the center. if this force is a short range force then its easy to show that the potential in the center of the mass is inversely proportional to the square of the radius of the orbital. on the other hand this force could be purely quantum mechanical and have no corresponding field. since the angular momentum of the antiparticles (eventually) cancel each other out completely(?) there is nothing to prevent it from collapsing to a point. (although doing so would release an infinite amount of energy so maybe the momentum doesnt cancel out completely)

the main thing to remember here is:
http://en.wikipedia.org/wiki/Positronium#Energy_levels
the 'energy' levels are approximately half those of the hydrogen atom. they are exactly half of what hydrogen would be if the proton had infinite mass. btw fwiw I think the second electron of helium is also exactly half the 'energy' of the first.

I really dont know what to make of all that.

to picture the field lines of a short range force imagine a flat 2 dimensional circle of charge in a 3 dimensional space. (inverse square is short range in a 2 dimensional world). the component of the field within the plane of the circle is virtually negligible everywhere except near the surface.(the field lines in the center being perpendicular to the circle).

the centrifugal force at the equator of an orbital with radius r is 1/r^3

btw I dont view this model as contradicting quantum mechanics. I view it as being simply a different interpretation of quantum mechanics.

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on the other hand, it could simply be (for an electron in the ground state of a hydrogen like atom) that if the mass is in an orbital with radius of one then 2/3 of the charge is in that same orbital while 1/3 the charge is in an orbital with radius of 2. that also gives a gyromagnetic ratio of 2 and might explain why the electron doesnt always seem to fall to the lowest orbital (2/3rds of the charge WOULD always fall to the lowest available orbital). the orbitals of positronium would have to be explained by some other mechanism, possibly just partial cancellation of angular momentum, though it seems strange that it follows the same pattern.

http://en.wikipedia.org/wiki/Hydrogen-like_atom#Non-relativistic_Wave_function_and_energy

Z is the charge on the nucleus

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You had better give up imaging the spin of the electron.
Is is abosolutely impossible.

To explain the spin g factor 2, and spin angular momentum 1/2 hbar,
the charge and mass of one electron must be separated and widely distributed.

But in the scattering experiment, one electron is too small to measure the size.
So the mass of one electron is not so widely distributed.

There is one more critical problem about the electron spin.
The spinning electron must be rotated by an angle of 4π in order to return to their original configuration.

We can never describe such a strange particle.