# Where does the electron get its high magnetic moment from?

I have always found the concept of spin a little weird. I had read somewhere that for the charge or size of electrons, their magnetic field is very high. In order to produce such fields, they must be spinning faster than the speed of light which is not possible. So where do the electrons get their high magnetic field from?

An answer without too much math would be preferred.

So I quote this source http://scienceblogs.com/principles/2010/07/26/electron-spin-for-toddlers/ :-

If we want to say that the magnetic moment of the electron is due to the motion of a spinning ball of charge, then we can easily calculate what the spin rate should be, given what we know about the size of an electron. If you use the maximum size you could possibly associate with the electron, the “classical electron radius”, and calculate how fast a sphere of that size would need to be spinning to produce the observed magnetic moment, you find that a point on the surface would need to be moving at a speed several times the speed of light in vacuum, which is impossible. That’s also a gross overestimate of the size of an electron– as far as well can tell, the electron has no physical size. It’s a point particle, and thus doesn’t have a surface that can be physically rotating.

OK, then, maybe the magnetic moment is just one of those things, you know? Maybe the “spin” angular momentum isn't really angular momentum at all. This is also false– spin angular momentum is real angular momentum. We know this because you can drive transitions from one spin state to another using polarized light, and we know from careful experiments done in 1936 that the angular momentum carried by light is real angular momentum. The angular momentum of a polarized photon can be used to make physical objects rotate, and it can also be used to make electron spins change states; this at least strongly suggests that spin angular momentum is real angular momentum comparable to that of spinning basketballs and all the rest.

So I do not understand what all this is suppose to mean. It first says maybe it is not spinning at all and then says that it is real angular momentum. Please clarify this for me. Or is this topic yet to be researched upon?

I think I have nearly got the answer but it was not what I was expecting (I am not talking about the quality of the answers). In the end I just want to ask as the electron actually does not spin but has angular momentum. I have pretty much understood that actual spinning is not the reason for its angular momentum. So is it safe to say that the actual reason behind the angular momentum is unknown?

First, the electron isn't actually spinning. Physical objects made up of collections of electrons and protons (and neutrons) can have angular momentum because they rotate; the electron does not get its angular momentum for the same reason.

Second, the magnetic moment of an object with angular momentum L is proportional to

$$\mu \propto \frac{qL}{M}$$

The angular momentum of an electron due to its spin is a fixed constant, proportional to $\hbar$. Since the electron has a very small mass, its magnetic moment is very large.

• But if the electron isn't actually spinning, then where does it get its angular momentum from? You have said it isn't actually spinning. So where exactly does it get the angular momentum? I am a little confused. I have read many sources. Some say it has an intrinsic spin and spins and some just say it doesn't actually spin without specifying anything else. Please help. Jul 18, 2014 at 5:19
• This is a complicated topic called "spin" which I won't attempt to discuss here. You can find discussions on it in almost any book on quantum mechanics, e.g. Sakurai. But the main point is that the electron just plain has angular momentum. It's a logical fallacy to think that just because something has angular momentum, it must have it for the same reasons as other things you're familiar with. Until Michelson-Morley / Einstein, people swore that electromagnetic waves needed an aether ... this is no different. You've learned one reason things can have angular momentum. It's not the only one. Jul 18, 2014 at 13:44
• So can I say there is a particular reason which is not actual spinning but it is yet to be discovered? Jul 18, 2014 at 14:44
• You can say "there is a particular reason which is not actual spinning." I don't know why think it hasn't been discovered. We have a coherent understanding of the intrinsic angular momentum of particles (misleadingly called "spin"). Of course our understanding could be wrong, and we may be yet to discover something (more) correct. But, it's not a mystery at the moment. Again, I refer you to a source like Sakurai or Griffiths QM textbooks. Jul 18, 2014 at 15:15
• I am only 15 and I cannot understand the complex math involved. But looks like you misinterpreted my second question. I meant the reason for that angular momentum. How does it get it and all? I know it is an intrinsic property like mass, but even mass is obtained from somewhere (the Higgs mechanism). So that is what I meant for an actual reason for that angular momentum. Jul 18, 2014 at 15:38

For what it's worth, I've always had the same feeling that the spin should have some sort of reason behind it. It seems so unsatisfying to be told more or less that "it just came that way." Is there any deeper sort of explanation at all? I recall a paper in AJP from years ago called "What is spin?" by Ohanian, but I didn't put in the effort to follow it. I remember it is referenced in a footnote in Griffiths, too.

Hey -- I found it--

http://www.physics.mcmaster.ca/phys3mm3/notes/whatisspin.pdf

This might be along the lines of what you are after.

Added after rereading Ohanian's paper -- A quick look through this paper has me convinced that the explanation he gives is exactly what you are seeking. I think the punch line comes across even if you don't bother to follow all the mathematical details.

As the previous post mentioned, forget about the concept that the electron is actually spinning. Spin, like rest mass and electric charge, is an intrinsic property of subatomic particles. Yes, it's angular momentum. No, nothing is spinning. Although many physicists today do not like this explanation, special relativity introduces a useful analogy with mass. A particle at rest in some reference frame begins with mass $m_0$. In another reference frame, it is moving with speed $v$. In this frame, the particle's mass is given by

$$m_{rel} = m_0 \frac{1}{\sqrt{1 - v^2/c^2}},$$

where $c$ is the speed of light in vacuum. Notice that there exists a frame where the particle has mass even though it is motionless. Although this analogy isn't perfect, one can draw upon similar reasoning to accept that particles can have intrinsic angular momentum.

To be honest, "spin" is a confusing and unfortunate word choice, because nothing is spinning. The world simply refers to the presence of angular momentum, like in a top (which is spinning, unlike the electron). Quantum mechanics has shown us that our world differs greatly than what we would expect from everyday life. While there are numerous analogies connecting quantum phenomena with familiar experiences, the fact is that the quantum world is different from the classical one we're used to assuming we live in. Analogies only go so far, and at the end of the day mathematics is the only way we can truly explain these phenomena.

Edit: I didn't exactly answer the question, but rather tried to explain the difference between intrinsic angular momentum and that due to physical motion. As the previous post also mentioned, the electron has such a high magnetic moment due to its small mass.

• But my question is where does that angular momentum come from. If it isn't arising from actual spinning, then from where? For example the mass is given by the Higgs Field. So I need an explanation for where does that angular momentum actually come from? Jul 18, 2014 at 5:51
• It's part of the wave function. It's intrinsic, no more given to it than are coordinates in space-time. It's a quantum thing, and you seem stuck on a classical picture. And the Higgs field isn't the only way to acquire mass. Jul 18, 2014 at 6:41
• @zibadawatimmy I know but there should be something at least that gives it angular momentum. To have some magnetic momentum, it should at least have some little spin. Jul 18, 2014 at 7:09
• @rahu You're still stuck in the classical picture. You envision the angular momentum as something acquired from something. It isn't here. That's why the word intrinsic is used. It has angular momentum in the same way that you have lungs: it's built into you. Jul 18, 2014 at 7:12
• Then what does that angular momentum correspond to? How does it look like when observed? Read this from what I have quoted:- Maybe the “spin” angular momentum isn’t really angular momentum at all. This is also false– spin angular momentum is real angular momentum. Jul 18, 2014 at 7:25

The electron spin can be associated to the electromagnetic field surrounding a free electron. Note that an electron not only has a charge e but a magnetic momentum (Bohr`s magneton $\mu_B= e/(2 m_e ) \hbar$). Thus anywhere in the space surrounding an electron except along its spin axis there exists a non-zero Poynting vector field $\mathbf S=\mathbf E \times \mathbf H$ corresponding to energy flux circulating around the electron. In terms of relativistic electrodynamics this energy flux corresponds to momentum flux and finally angular momentum. This electromagnetic field angular momentum can equate the observable spin angular momentum $\hbar/2$.

(See the following links: “The electron point model”: http://www.gy.com/point.html and “Singularity-free electrodynamics for point charges and dipoles: a classical model for electron self-energy and spin” by S.M. Blinder: http://arxiv.org/abs/physics/0208072 Hence electromagnetic angular momentum is not necessarily bound to any kind of classical rotation.

In contrast, imagine a charged pellet in rotation. As its charge co-rotates with its atoms it will generate a magnetic dipole field intermeshing with the electrostatic field of the pellet in such a way that electromagnetic field angular momentum is generated, in addition to the classical angular momentum of the pellet.

• Note that this site has MathJax enabled, which means you can use Latex-like syntax to add in equations for readability. I've added them in for you here, but hopefully you'll be able to follow it along for later use! Dec 31, 2015 at 13:32