The idea of quantum particles possessing some kind of intrinsic angular momentum is a pretty weird one, but it turns out to be necessary to understand a huge amount of physics. That intrinsic angular momentum is called "spin", but it's *not* correct to think of it as resulting from the particle being an extended physical object actually spinning. As I learned from reading The Story of Spin (cool book by Tomonaga, though I found it a bit impenetrable toward the end - more on that below), Kronig first suggested that electrons might have intrinsic angular momentum and used the intuitive idea of spinning to describe it; Pauli pushed back very hard on Kronig about the idea that there could be some physical rotational motion involved - the intrinsic angular momentum is some constant on the order of \(\hbar\). If it were the usual mechanical motion, dimensionally this would have to go something like \(m r v\), where \(m\) is the mass, \(r\) is the size of the particle, and \(v\) is a speed; as \(r\) gets small, like even approaching a scale we know to be much larger than any intrinsic size of the electron, \(v\) would exceed \(c\), the speed of light. Pauli pounded on Kronig hard enough that Kronig didn't publish his ideas, and two years later Goudsmit and Uhlenbeck established intrinsic angular momentum, calling it "spin".

Because of its weird intrinsic nature, when we teach undergrads about spin, we often don't emphasize that it is just as much angular momentum as the classical mechanical kind. If you somehow do something to a system a bunch of spins, that can have mechanical consequences. I've written about one example before, a thought experiment described by Feynman and approximately implemented in micromechanical devices. A related concept is the Einstein-de Haas effect, where flipping spins again exerts some kind of mechanical torque. A new preprint on the arxiv shows a cool implementation of this, using ultrafast laser pulses to demagnetize a ferromagnetic material. The sudden change of the spin angular momentum of the electrons results, through coupling to the atoms, in the launching of a mechanical shear wave as the angular momentum is dumped into the lattice. The wave is then detected by time-resolved x-ray measurements. Pretty cool!

(The part of Tomonaga's book that was hard for me to appreciate deals with the spin-statistics theorem, the quantum field theory statement that fermions have spins that are half-integer multiples of \(\hbar\) while bosons have spins that are integer multiples. There is a claim that even Feynman could not come up with a good undergrad-level explanation of the argument. Have any of my readers every come across a clear, accessible hand-wave proof of the spin-statistics theorem?)

## 7 comments:

Sudarshan and Duck have an Am. J. Phys paper about this [Am. J. Phys. 66(4), 284 (1998)], which discusses this in some detail.

Sir Michael Berry and J. M. Robbins use non-relativistic arguments to justify the spin-statistics connection.

https://michaelberryphysics.files.wordpress.com/2013/07/berry286.pdf

'Cept it's not true in non-relativistic quantum mechanics. There's nothing inconsistent about spinless fermions, or spin-1/2 bosons in non-relativistic quantum mechanics.

I don't know how to avoid, at some stage, invoking the CPT Theorem (which, in turn, involves the fact that the Euclidean continuation of the Lorentz group is connected). So I don't see an elementary proof.

"we often don't emphasize that it is just as much angular momentum as the classical mechanical kind"

But it should not be emphasised too much because they are fundamentally different. Essentially spin 'looks like' classical angular momentum because of a mathematical coincidence involving the groups SO(3) and SU(2).

StevenG, I know what you're saying from the formalism perspective, but given that they can be combined (e.g., J = L + S), how critical is that distinction to measurable phenomena in the lab?

Douglas Natelson: I think it isn't critical as far as measurement is concerned. I just wanted to point out the different natures. I think of it as similar to how you can add an electromagnetic force and gravitational force on an object to get the total force. They both look like forces but have a different theoretical and physical origin.

I think it's worth stating that the spin-statistics theorem is only really a theorem for free field theories. For interacting systems, it isn't provable. I think this is just a mathematical physics issue, as it really seems that the theorem always holds, but who knows!

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