Spin measurement

[Raphanus]

Is spin something one can measure in the following sense:

Is there a Hermitian operator S which when applied to a multi-
component wave function \psi gives the following?

S\psi = s\psi, where s is spin?

Or is spin defined by the transformation properties of \psi ?

That is \psi* = S(\psi)S*; SS*=1

and from looking at S one call say what s is? If so, how is that done?

[Raphanus]

On Sep 3, 11:57 pm, "Jim Heckman" <rot13(reply-to)@none.invalid>
wrote:
> On 31-Aug-2008, Raphanus <lester.we...@gmail.com>
> wrote in message
> <21a11003-5d21-473f-9abc-6bc3002a8...@2g2000hsn.googlegroups.com>:

>
> > Is spin something one can measure in the following sense:
>
> > Is there a Hermitian operator S which when applied to a multi-
> > component wave function \psi gives the following?
>
> > S\psi = s\psi, where s is spin?
>

> Sure, if \psi is an eigenstate of S.  For example, an electron with
> its spin pointing upward along the z-axis is in an eigenstate, with
> eigenvalue +1/2 h-bar, of the Hermitian S_z operator.
>
> And of course, as with any observable, a physical measurement of a
> spin component of a quantum state must result in an eigenvalue of
> the spin operator, after which the state will "collapse" to its
> projection onto the measurement result's eigen-subspace of the
> relevant Hilbert space.

>
> > Or is spin defined by the transformation properties of \psi ?
>
> > That is \psi* = S(\psi)S*;   SS*=1
>

> I don't understand what you're saying here.  What is true is that
> any spin is, by definition, an angular momentum, which is to say
> that it satisfies commutator relations [S_x, S_y] = i*h-bar*S_z,
> [S_y, S_z] = i*h-bar*S_x and [S_z, S_x] = i*h-bar*S_y, and is a
> representation of the Lie algebra so(3) = su(2) and thus generates
> a representation of the rotation group SO(3) (or of its
> simply-connected double cover Spin(3) = SU(2)).

>
> > and from looking at S one call say what s is?  If so, how is that done?
>

> --
> Jim Heckman

Thanks Jim. Your answer helps. Maybe I can clarify the source of my
confusion by rephrasing my question

Suppose \psi = (a,b,c,d) - a four component wave function (a,b,c,d in
C). I'd like to know if it is has total spin 0, 1/2 ,1, ...etc. with
no other knowledge than (a,b,c,d). Is this possible? I think I
understand that I can find the z-component if I know it is 1/2, but my
question more concerns "how do I know it's 1/2"?

Thanks again for your time and attention.