Can comeone explain the difference between spin 1/2, spin 1 and spin 2 in such a way that the numbers make sense. I realize they mean quantum spin and relate to classic spin by analogy only but in what sense is spin 2 "twice" spin 1, or spin 1 "twice" spin 1/2. I realize also that spin 1 and 2 are in a different "family" than spin 1/2.

OK, first you need to know that when someone calls an electron a "spin-1/2 particle", they don't mean that its spin angular momentum has a magnitude of 1/2. What they mean is that the z-component of its spin angular momentum has a magntude of 1/2 times h-bar.

 

Start with the spin angular momentum vector:

 

[math]

\mathbf {S}=S_x \mathbf {i}+S_y \mathbf {j} +S_z \mathbf{k}

[/math]

 

In QM, the components of angular momentum do not commute, which means that you cannot simultaneously measure all 3 components. You can, however, measure the magnitude of the angular momentum and a single component simultaneously. Let's call this single component "the z component". We have the following relations:

 

[math]

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

[/math]

[math]

S_z=m_s\hbar

[/math]

 

Here, m_s is the spin quantum number and s is the number from which you calculate the norm of the spin vector (I don't know if it has a special name or not). The quantum number m_s can take on any values in the set {-s, -s+1, -s+2,...,s-2,s-1,s}.

 

The number s is what is referred to when people say "spin (insert number here)". So for instance a spin-1/2 particle has s=1/2, which means that the magnitude of its spin angular momentum vector is given by the following.

 

[math]

|\mathbf {S}|=\frac {\sqrt {3}}{2}\hbar

[/math]

Spin refers to the amount of intrinsic angular momentum, in terms of h-bar, a particle has. The amount refers to the vector projected along some axis (usually the z axis by convention). The angular momentum projection can only change by units of h-bar. This picture might help

 

A particle that is spin 1/2 will have 1/2 h-bar of angular momentum projected along the z axis. That can be "up" or "down," i.e. +1/2 or -1/2 in terms of some defined axis.

 

A spin 1 particle can have 1,0 or -1 units projected along the z axis.

 

Two spin 1/2 particles may combine to give either a spin 0 particle (anti-aligned) or a spin 1 particle (aligned spins)

Thank-you both and that link looks interesting.