Undergrad Does the electron really spin 720 degrees?

[Mark Harder]

It seems to me that the semantics is confusing everyone but the true experts, who freely manipulate the quantum variable called 'spin' and don't particularly worry about what that term means beyond that of a certain mathematical construct in the language of Hilbert space, operators and state functions and so forth . The rest of us are scratching our heads and wondering how an object can possesses a 'spin' and not actually spin. It's a fair response, you must admit. I'm grateful to whoever, was it Glashow?, coined the term 'quark'. Better to invent a nonsense term for something that makes no sense to pedestrians like me. IMHO, if you're going to say that a quantum object has no analogy in the macroscopic world, then don't use an macroscopic analogy like spin.
Is spin an object with phase in the sense that periodic functions can differ by some fraction of their period and return to ? In particular, if the phase changes by one period, the function is indistinguishable from the original. The function can be said to be transformed by elements of a Group, which must contain a transformation, the identity element of the Group, that leaves the object in its original state. That's a classical analogy, dealing with functions and groups. The quantum equivalent would be a state vector with 'spin' whose phase is transformed by an operator. That's strictly guesswork on my part, drawn from the attempts to come to terms with the concept that I read here. Feel free to correct, politely.

Perhaps we here in PhysicsForums should have a contest to see who comes up with the best substitute for quantum "spin". First word I came up with was "luck". Any takers?

 

[vanhees71]

You don't need to rename well-established and well defined things. Gell-Mann had the right to invent a name for "quarks", because he was the one who discovered them (first thinking of them as if they were just purely mathematical auxilliary constructs, and it took Feynman to convince him and the rest of the community of their "reality" in terms of his parton model of hadrons to describe deep-inelastic scattering).

The spin is a quantum concept. You cannot describe it in any other way than with quantum theoretical means, and the most clear way is to use the analysis of the (covering of) the rotation group, SU(2).

 

[Collin237]

Spin is not analogous to classical rotation, but it's convertible with it. They are both parts of the same conserved quantity, angular momentum, which is the Noether current of rotational symmetry. This was confirmed in 1910 by Einstein and de Hass in an actual laboratory experiment in 1910, before the math now known to correctly describe spin had even been proposed!

A word like "quark", as you said, appeared through the social act of naming, so it can be attributed to (or, depending on one's tastes, blamed on) the person who coined it. But discovering a phenomenon that only much later becomes a concept, as was the case with spin, is beyond the capability of society. So a fair description of spin cannot gloss over the natural justification for calling it what it demonstrably is.

 

[rrogers]

It seems to me that "spin" in this case can be mathematically modeled by a fiber extension to a manifold and then a nontrivial connection. Is this correct and can somebody give a reference (hopefully open-source) I can read?
As a possible separate question (?): I have mused that a standard physical "image rotator" is analogous to a spin 2 system. That is, the image rotates twice as fast as the rotator; this results from a rotating axis of inversion/mirror.

 

[Collin237]

What is an "image rotator"? I googled that phrase and all I could find was ads for a gallery carousel widget. I don't think that's what you're referring to.

If you mean a rotor (a quaternion interpreted geometrically), that's what I was referring to in #52. The ratio of rotations works the other way, and gives a rotor a spin of 1/2.

 

[rrogers]

What is an "image rotator"? I googled that phrase and all I could find was ads for a gallery carousel widget. I don't think that's what you're referring to.

If you mean a rotor (a quaternion interpreted geometrically), that's what I was referring to in #52. The ratio of rotations works the other way, and gives a rotor a spin of 1/2.

If the hall monitors move this; please leave a pointer since I think the material is worth knowing. In any case, a comparison with "spin 2" theory would be interesting; or if it's irrelevant I would like to know that as well.
All of the references below have the same thing in common: imagine a tube to look through, then imagine a line across the tube that is an axis of reversal, now rotate the line of reversal. Since the line has the same effect when in the "normal" position and when it's rotated 180 degC; the image must travel twice as fast. When I understood this while working on some optics I went to the tool shed and found a solid tube of clear plastic. Sure enough when you position your eye behind the tube so that the image is reversed (say left to right) then rotating the tube causes the viewed scene to travel twice as fast.
On a more professional level:
Here is a prism type:
https://en.wikipedia.org/wiki/Schmidt–Pechan_prism
Here is a mirror references: first wikipedia quote
"In the case of two mirrors, in planes at an angle α, looking through both from the sector which is the intersection of the two halfspaces, is like looking at a version of the world rotated by an angle of 2α"
https://en.wikipedia.org/wiki/Mirror_image
Or if you are a technical masochist (like myself):
https://www.colgate.edu/portaldata/imagegallerywww/98c178dc-7e5b-4a04-b0a1-a73abf7f13d5/ImageGallery/geometric-phase-of-optical-rotators.pdf
I haven't read it yet but it looks like my understanding.

 

[rrogers]

If the hall monitors move this; please leave a pointer since I think the material is worth knowing. In any case, a comparison with "spin 2" theory would be interesting; or if it's irrelevant I would like to know that as well.
All of the references below have the same thing in common: imagine a tube to look through, then imagine a line across the tube that is an axis of reversal, now rotate the line of reversal. Since the line has the same effect when in the "normal" position and when it's rotated 180 degC; the image must travel twice as fast. When I understood this while working on some optics I went to the tool shed and found a solid tube of clear plastic. Sure enough when you position your eye behind the tube so that the image is reversed (say left to right) then rotating the tube causes the viewed scene to travel twice as fast.
On a more professional level:
Here is a prism type:
https://en.wikipedia.org/wiki/Schmidt–Pechan_prism
Here is a mirror references: first wikipedia quote
"In the case of two mirrors, in planes at an angle α, looking through both from the sector which is the intersection of the two halfspaces, is like looking at a version of the world rotated by an angle of 2α"
https://en.wikipedia.org/wiki/Mirror_image
Or if you are a technical masochist (like myself):
https://www.colgate.edu/portaldata/imagegallerywww/98c178dc-7e5b-4a04-b0a1-a73abf7f13d5/ImageGallery/geometric-phase-of-optical-rotators.pdf
I haven't read it yet but it looks like my understanding.

Actually you can model spin 1/2 by attaching an arrowhead to the reversal line that is invisible to the observer but visible to a second observer. The second observer could see the arrowhead. The second observer would say that the first observer has to rotate the line/image twice to get back to the original "state"; even though the first observer would see no difference. BTW: I think I can formulate the whole process in matrix form; but haven't done it yet.

 

[yoron]

Here's a simple answer. If you consider a large object like the Earth, it has orbital angular momentum (from its orbit round the Sun) and spin angular momentum from its rotation about its own axis. But, these two are physically the same: the spin angular momentum of the Earth is just the orbital angular momentum of all the particles that make up the Earth as they rotate about the axis.

The spin angular momentum of an electron, however, is essentially different from its orbital angular momentum. It is NOT the orbital angular momentum of all the stuff that makes up an electron as it spins on its axis.

The electron's spin does, however, share mathematical properties with orbital angular momentum, but it's a fundamentally different brand of angular momentum from anything we see around us.

Lovely answer. One might want to add that a electron also can be defined as a 'standing wave' around some nucleus to show how, well, ridiculous it becomes to call it a equivalence to a classical spin. At least from where I look at it.

 

[vanhees71]

No, that's also not an adequate picture of an electron, which is an electron; full stop. To our present knowledge it's an elementary particle described by a quantized Dirac spinor field in the Standard Model of elementary-particle physics. You cannot say anything else about it than that. For sure it's neither a classical point particle (which always is a macroscopic object whose extension is irrelevant for the situation to describe, i.e., it's sufficient to consider only its center of momentum and in this sense idealize it to a point obeying the laws of (relativistic) classical mechanics) nor a classical wave.

 

[yoron]

Heh, depends on where you look at it from, doesn't it?
Whatever a 'electron' now might be :)

 

[vanhees71]

Though shalt not make (inadequate) images! ;-)

 

[yoron]

Though shalt not make (inadequate) images! ;-)

try a 'field' :)