Understanding Quantum Spin Number vs Angular Momentum

[tashi]

hi,
i am trying to work out exactly what is the difference in quantum mechanics between
1) spin, as in 'angular momentum' and
2) spin as in 'quantum spin number'.

It seems like quantum spin number only describes certain properties, like whether a particle is a fermion or a boson (so it wouldn't make any difference if we labelled different 'spins' 'apples' and 'pears' rather than '1,2,3' or '1½,2½,3½..') while angular momentum seems to be a relative physical quantity -- i.e. a particle with angular momentum, say 3, has 'higher' angular momentum than one with angular momentum 1, and one with angular momentum 2 is somewhere imbetween. Is that correct?

From what I've read some sources seem to say these two values are independent. Elsewhere I've heard that quantum 'spin number' is actually just the z-component of the angular momentum of the particle, and the spin number is a way of describing the way the particle is actually physically spinning. Any clues?

 

[inha]

Spin has nothing to do with actually physical spinning. And it is not the z-component of the angular momentum L. What the folks elsewhere told you are not correct.

 

[tashi]

so what is spin?

 

[inha]

What it really is? I have no idea. Since you're asking this question I take it that intrisic angular momentum probably isn't a sufficient answer but that's the best one I can give. I don't know what mass or charge really are either but they are properities of particles too.

 

[HallsofIvy]

The problem is that in the "micro-world", electrons, muons, etc., our basic concepts of size, shape and all just don't work. An electron is NOT a little ball and it simply makes no sense to talk about it "spinning". The "spin" of an electron is a property that appears to be important (for example, in distinguishing between fermions and bosons). It is called spin because spins happen to combine "vectorially" in the way that angular momentum does in the "macro-world". There is a completely different property that combines in the same way and is called "iso-spin". Yet another property seems to have 3 "basis" components much the way our eyes see colors in terms of Red, Blue, Green and, so, is referred to as "color". I'm sure you understand that the "charm" of an elementary particle has nothing to do with our usual meaning of the word "charm"!

 

[tashi]

thanks guys.
so do electrons have angular momentum then?
if so, is this different from their quantum spin number?

 

[inha]

Yes and yes. Oh and when you say spin quantum number you should just say spin because even though the quantum number contains information about the spin state it's still just a number.

 

[jtbell]

The "spin" quantum numbers $s$ and $m_s$ are related to the allowed values of the magnitude and z-component of the intrinsic angular momentum of a particle:

$$S = \sqrt{s(s+1)} \hbar$$

$$S_z = m_s\hbar$$

For an electron, with $s = 1/2$ and $m_s = \pm 1/2$, these give

$$S = \frac {\sqrt{3}}{2} \hbar$$

$$S_z = \pm \frac{1}{2} \hbar$$

Although an electron is not a little tiny ball spinning around its own axis, $S$ and $S_z$ do, neverthess, specify real, physical angular momentum which can be added to macroscopic angular momentum, and which figures into conservation of total angular momentum of a macroscopic system. This has been observed experimentally, in the Einstein-de Haas effect. (try a google search on both "einstein de haas effect" and "einstein dehaas effect")

 

[dextercioby]

The quantum theory of angular momentum is nothing else but representation theory of $SU(2)/su(2)$ in the context of Wigner's theorem, that is: one's searching only for unitary symmetry group representations.

The electron's spin angular momentum is found elegantly by deducing the $\frac{v^{2}}{c^{2}}$ approximation of Dirac's equation.

Daniel.