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I'm currently reading Alonso and Finn's Electromagnetism book.

It explains that the spin contributes to the magnetic moment and is somewhat comparable to a rotation of the particle around its own axis. It says that the spin of a particle is caused by a certain internal structure, which makes sense in the aforementioned analogy.

Right underneath the paragraph with the explanation of spin, it says "The electron has no known internal structure", but since it does have a spin, does that mean that we know the electron has an internal structure but we just don't know what it is?

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No, it means that as far as we know, the electron has no internal structure. –  Javier Badia Aug 12 at 12:56
    
But then were does the spin come from? –  Joshua Aug 12 at 12:57
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@Joshua Spin cannot be understood classically! This is an important, but difficult fact that everyone has to simply live with. –  Danu Aug 12 at 13:39
    
Possible duplicates: physics.stackexchange.com/q/1/2451 , physics.stackexchange.com/q/822/2451 and links therein. –  Qmechanic Aug 12 at 21:37
    
Aside from the fact that we have no positive empirical evidence for internal structure in the electron, there is an additional issue, which is the "confinement problem." If the hypothetical sub-parts of an electron (called preons) are confined to a space of size x, then the uncertainty principle says the mass-energy is at least about h/x. This would make the preons more massive than the electron they supposedly make up. The confinement problem can be worked around in some cases, but it's a reason not to expect to see substructure inside electrons. –  Ben Crowell Aug 12 at 22:30

3 Answers 3

up vote 14 down vote accepted

Spin is not about stuff spinning. (Confusing, I know, but physicists have never been great at naming things. Exhibit A: Quarks.)

Spin is a purely quantum mechanical phenomenon, it cannot be understood with classical physics alone, and every analogy will break down. It has also, intrinsically, nothing to do with any kind of internal structure.

(Non-relativistic) spin arises simply because quantum things must transform in some representation of the rotation group $\mathrm{SO}(3)$ in order for the operators of angular momentum to act upon them (and because we need to explain the degree of freedom observed in, e.g., the Stern-Gerlach experiment. Since the states in the QM space of states are only determined up to rays, we seek a projective representation upon the space, and this means that we actually represent the covering group $\mathrm{SU}(2)$. The $\mathrm{SU}(2)$ representations are labeled by a number $s \in \mathbb{N} \vee s \in \mathbb{N} + \frac{1}{2}$, which we call spin. Whether the thing we are looking at is "composite" or "fundamental" has no impact on the general form of this argument.

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I tell myself all the time that I want to learn a bit of quantum mechanics. Then I see questions/answers like this, and I get completely discouraged. –  Cruncher Aug 12 at 15:24
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@Cruncher: Noooo, don't get discouraged. Is it the bit of group theory in here that caused that reflex? It's a beautiful bit of math, and, if taught well, not harder than any basic analysis or linear algebra concepts. And it is not like you must delve into this stuff directly from the beginning. Much of intro QM does not need spin. –  ACuriousMind Aug 12 at 15:32
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@Cruncher You can't understand Bell's theorem without having at least an introduction to QM like Auletta's book. –  metacompactness Aug 12 at 15:46
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@Cruncher Or, if you don't like much mathematics, try Feynman's lecture (the last volume is QM). –  metacompactness Aug 12 at 15:48
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@Zack: You're correct that spin is, essentially, angular momentum - but that does not mean that anything is actually spinning, which is the strange thing about spin. –  ACuriousMind Aug 12 at 16:55

"The electron has no known internal structure", but since it does have a spin, does that mean that we know the electron has an internal structure but we just don't know what it is?

An electron has no known internal structure simply means that nobody knows if the electron has an internal structure. So far they know none and therefore they suppose it has none

Spin is not related to an internal structure. If you consider the electron as the classical ball, the ball can spin both with or without an internal structure.

But spin is now considered an intrinsic property of the electron, which means that the effects are those of a classical spin, but the particle must not necessarily spin.

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Similarly, a classical ball with an unremarkable surface will appear identical to a similar ball which has some angular momentum, though it defies visual observation. –  KidElephant Aug 12 at 15:07

Spin is a wave property. It exists in classical relativistic wave theories as well. A circularly polarized wave carries an angular momentum that's related to the spin of the field. A gravitational wave (spin-2) can carry twice the angular momentum of a classical electromagnetic wave (spin-1).

Being "pointlike" is a particle property. You can think of the field value at a point as being related to the presence of a particle there. If the field's associated particle is an extended object (like a pion) then it doesn't just occupy the point where the field is nonzero, but also nearby points, which means that the interaction with a pointlike test particle depends not only on the field value at the test particle's location, but also on nearby field values. If the field's particle is pointlike (like the photon) then the force depends only on the field value at that point. Even classically, you could say that the electromagnetic field is "pointlike" since the Lorentz force only depends on the field at a point, though the terminology makes less sense without wave-particle duality.

So while spin is certainly a property of the particle (inasmuch as the particle and the wave are the same thing), it's not a property that depends on any internal structure of the particle.

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