I recently read one of those introduction books on physics and was puzzled by one of its illustrations. In the chapter explaining an electron's spin, the electron was illustrated like a sphere with a line drawn across the middle with arrows indicating left-to-right. A line is also drawn from the bottom of the sphere to the top with an arrow indicating "up". This drawing was supposed to illustrate an electron's spin-up which spins clockwise. But isn't left-to-right counter-clockwise? Is the illustration mistaken?
quote: But isn't left-to-right counter-clockwise? Is the illustration mistaken? =============================================== Yes, that would be correct. Possibly a mirror image of the illustration was published in the book. What I have always wondered though, is how "up" is distinguised from "down" in a rotating sphere. If looked at from the "down" pole, a sphere's rotation is the opposite of what would appear if looked at from the "up" pole. In other words, how is the "up" or "north" end determined on rotating spheres?
Spin is not really a measure of the direction the particle spins in. That is more angular momentum. But it is one way of visualising what it is. Spin a quantum number characterising one possible eneergy state of a particle. An electron can be spin up or spin down (+1/2 or -1/2) but that does not stipulate direction of spin. It's just a handy phrase to use rather than inventing a new phrase such as spin-flibble or spin-flobble.
So when someone says an electron with spin-up spins clockwise they don't mean it literally? What then do they actually mean? And the clockwise/anti-clockwise phrase - is it from the perspective of the electron or from the perspective of an observer observer the electron? Also, I was told that one way to remember the spin is by the thumbs-up gesture - the right fist clenched with the thumb sticking up. The thumb indicates up/down while the clenched fingers indicates the clockwise/anti-clockwise spin. Is this true? If it is, then I'm even more confused. The clenched fingers of the right fist - with the thumbs pointing up, is also clenched from right-to-left, therefore counter-clockwise.
I am not the one that is the most competent about spin staff, and probably I will stuck you even more , but I have to share something that bothers me about this explanation of the spin as some kind of rotation. Spin have nothing to do with any kind of rotation! You see, we have the upper limitation of velocity which is c. But if electron starts to spins around its main axis it should take some time, in another words particles with spin will never be able to rich velocity of c. Its rotation will slow them up according to the low for relativistic sumation of the velocities. In even simlier words electron rotation will eats its forward speed because all movements that electron makes should fit in velocity of c. Also why: 1) all electrons have spin either 1/2 or -1/2 , that means that speed of rotation is equal for all of them, but velocity is not discreat , it can have continuous values. There is nothing that to explain why all the electrons rotates with the same angle velocity. 2) there is many axis of rotation, how happens so that all electrons choose one and the same axis?
1. i don t agree with the people who say that spin has nothing to do with rotation, or that the "up" in "spin up" doesn t mean up. spin has quite a lot to do with rotation, just don t make the mistake of thinking of the electron as a sphere with finite radius. this angular momentum is all intrinsic. 2. although the electron cannot be said to be rotating in any meaningful sense insofar as thinking of spin as a rotating 3d body, spin up corresponds to counterclockwise rotation, and spin down to clockwise rotation, with the right handed orientation of the x-, y-, and z-axes.
I'll give you that. Spin is about spin but not in a classical sense. I hope you agree that the quantum number 's' has no classical analog. This is one of those horrible things that only makes sense in the maths. As the OP says, what is it spinning with respect to? How do you differentiate between Bose-Einstein an Fermi-Einstein stats by referring to the direction of spin? Last I saw there is no reference to direction in the maths, it's purely an eigenstate, not a vector.
Even though, as you say, spin has no classical analog, you still could have, in principle, classical fields with spin 1/2. What I mean is that classicaly, you have scalar fields, vector fields, etc. In principle, you could have a 2-component field that under rotation transform according to the 2-dimensionnal representation of the rotation group.
there is no 2 dimensional representation of the rotation group. thus you can have fields with spin 0 or spin 1, but not spin 1/2
The direction of spin is in the math, and spin is every bit as much of a (pseudo)vector as orbital angular momentum. While it is true that the state |+> is an eigenstate in an abstract Hilbert space, it is also the case that S is a pseudovector in regular configuration space, just like L. Very much to the contrary, angular momentum is the generator of rotations. And when L=0, spin is the generator of rotations. edit: fixed quote bracket
lethe, just wanted to be sure: spin-up corresponds to counter-clockwise rotation spin-down corresponds to clockwise rotation Is this correct? This is what I first thought when I saw the illustration but the accompanying text on the book put it as the other way round. Wish I could re-produce that part of the book here without infringing on any copyright law. One question though - what does right handed orientation refer to?
spin up means that the z-component of spin is along the positive z-axis. the positive z-axis is determined by choosing the x-axis and y-axis, and then to choose the z-axis you use your right hand: point your pointy finger in the direction you chose for x, then your middle finger in the direction you chose for y, and your thumb will point in the direction for z. if you use your left hand, you will get a set of axes with the opposite orientation, and then counterclockwise rotation (from positive x to positive y) would have an axis along negative z, and it would be spin down. so the sign on the spin vector depends on which choice of orientation for your axes you make.
Shame on me, you are right. The spin 1/2 representation is the 2D representation of SU(2) and not O(3).
the difference between quantum theory and classical theory is that states in a quantum theory are represented by rays in hilbert space, which means that adding a complex phase to the vector does not change the physical state. whereas in a classical theory, physical observables are represented by real numbers. therefore, in a quantum theory, you can have projective representations of the rotation group, which means it represents the group "up to phase". it so happens that every projective representation corresponds to a representation of the universal cover of the group. SU(2) is the universal cover of SO(3). thus, the reason we talk about representations of SU(2) instead of SO(3) is a purely quantum phenomenon. you have a 2d projective representation of SO(3), but the only way that it can act as a symmetry on a system is if the phase is not measurable, as in quantum theory.