1. I have a few questions for you, I'll ask one question at a time:

I understand that an electron in an atom spins like the wheels of a rollercoaster car: in one direction (parallel ) when the car is above the rail and in the opposite direction (antiparallel) when the car is upside down, as you know at some parks the track is actually called 'electron spin'. Is this metaphor appropriate?
I have read also that it moves in the orbit like on a Moebius strip, if this is true it means that it is always rotating around the imaginary rail, is it so?. If it is what makes the electron turn around the orbit?
Thanks

2.

3. You should not use the term electron spin, as that refers to something completely different. From your description, you are referring to orbitals.

4. Originally Posted by MeteorWayne
You should not use the term electron spin, as that refers to something completely different. From your description, you are referring to orbitals.
I am referring to electron spin in its orbitals

5. If you are actually talking about electron spin, then it absolutely does not spin like the wheels of a roller coaster. Spin in this context refers to something more abstract, the word spin is used more as an analogy, similar to the use of the word colour in chromodynamics.

6. Actually, I think you slightly overstate your case, my friend.

Spin describes a certain sort of continuous symmetry. These symmetries are in general described by Lie groups, and in this case by the group that goes by the catchy name of SU(2).

A typical element of SU(2) looks like for some real .

This "looks like" a rotation in , which we can think of as a mapping of into itself.. Since is a vector space, this is called a "representation" of our group.

Now any Lie group admits of at least 2 representations, one of which generates its algebra - let's park that for now. Our group SU(2) admits of 2 other representations, to each of which one can assign a rational number index, say, .

So when is an integer, one calls this a representation that describes the spin of "particles" called bosons, when for some odd integer one calls this a fermion, a particle with half-integer spin. The electron is a fermion with spin that is has spin one half.

You should note, this argument is all ass-backwards, and moreover the only known bosons have spin 1, although the conjectured Higgs boson is assumed to have spin 2

PS Here is another way of looking at our symmetries: SU(2) is a Lie grout and therefore a manifold, with the topology of a 3-sphere. In the spin 1 representation, for any point on this 3-sphere, whereas in the one-half spin representation . In other words you have to "rotate" twice before getting "home"

7. Do you have a good recommendation for a text on representation theory? If I am even ready for it - I have done four honours classes in real analysis, a couple honours algebra courses, I have taken a 300 level group theory course and I do know some complex analysis, as well as a few other less relevant math courses - if you think that is an acceptable prereq I would really like to do some reading on representations this summer.

In my group theory course we never covered lie groups formally, although we did look at SU(2) and a couple other groups which I think probably are lie groups. Basically I understood that last post, but that is around the limit of mathematical ability. Anyways, any resources on the subject you have would be greatly appreciated.

PS. So are all lie groups manifolds? I guess that makes sense since they if they are continuous. We showed that U(n) was a manifold in my last real analysis course while we were covering manifolds just as a side thing, that is interesting to know that all lie groups are manifolds.

8. Originally Posted by TheObserver
Do you have a good recommendation for a text on representation theory?
I hate recommending texts - obviously I don't know them all, and each person has a different way of learning.

That said, I have Fulton & Harris: Representation Theory, a First course. It is a graduate text, and very hard in parts. Plus it ain't cheap - I think I payed in excess of 30 euro several years ago
PS. So are all lie groups manifolds?
Most definitely, it is part of the definition!

Somewhere on my hard drive I have an essay on the matrix Lie group, which I can try and find if you (or anyone else, for that matter) would be interested. Best done in Math, though

9. Originally Posted by TheObserver
If you are actually talking about electron spin, then it absolutely does not spin like the wheels of a roller coaster. Spin in this context refers to something more abstract, the word spin is used more as an analogy, similar to the use of the word colour in chromodynamics.
The solved the problem of angular momentum saying that it is an intrinsic property of the electron, fair enough! Even though I suppose that if it were intrinsic it should always be present, and could never disappear, but...
If it doesn't spin at all, why the magnetic moment, and, how can the value be +, - 1/2 ? Only the the direction (parallel, antiparallel) can give opposite values. right?

10. Originally Posted by Guitarist
...whereas in the one-half spin representation . In other words you have to "rotate" twice before getting "home"
As per the moebius strip. Got a calculator? Work out . Don't worry about the dimensionality.

11. Originally Posted by Guitarist

PS Here is another way of looking at our symmetries: SU(2) is a Lie grout and therefore a manifold, with the topology of a 3-sphere. In the spin 1 representation, for any point on this 3-sphere, whereas in the one-half spin representation . In other words you have to "rotate" twice before getting "home"
So, not exactly like the wheels of a roller coaster. :-)

It's always hard to try and imagine the spin of particles. In one area of QM, particles are treated like point sources, in others like 3-spheres, while trying to imagine spin can only produce mental analogues of spin or no spin and one symmetry (for me anyway). Are there "directions" to the spin, or is there a possibility that a half spin is half a cycle of an oscillation? Or is it just one of those cases where trying to imagine it will fall short of what's really going on?

12. As i see it , the particle spin in super Position , it goes back and forward many time in time so what you gets is some average of the option and it stabilized at the present . Something like that thanks

13. Originally Posted by KALSTER
Are there "directions" to the spin, or is there a possibility that a half spin is half a cycle of an oscillation? Or is it just one of those cases where trying to imagine it will fall short of what's really going on?
There are directions to the spin angular momentum, either with or against the direction in which spin angular momentum is being measured. (positive or negative so to speak)

14. Originally Posted by wallaby
Originally Posted by KALSTER
Are there "directions" to the spin, or is there a possibility that a half spin is half a cycle of an oscillation? Or is it just one of those cases where trying to imagine it will fall short of what's really going on?
There are directions to the spin angular momentum, either with or against the direction in which spin angular momentum is being measured. (positive or negative so to speak)
As in an electron and a positron? Is that part of the reason they annihilate?

15. Originally Posted by KALSTER
As in an electron and a positron? Is that part of the reason they annihilate?
Nope, nothing to do with antimatter. (in fact i'm not sure what makes a particle an antiparticle) Basically what i was trying to say was, yes there are directions to spin. However since quantum mechanical spin has no analogue in classical mechanics i find it more useful to instead talk about the orientation of a particles angular momentum or magnetic dipole moment. (since these are the quantities that we can actually measure) I found a good way to learn about spin was by reading about the Stern-Gerlach Experiment.

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