[QM] Expectation value in spin-1/2 state

[Lengalicious]

Homework Statement

Basically I need to produce a state for a spin-1/2 particle such that the expectation value of <J_z> = 0 where <J_z> is for a spin-1 particle.

Homework Equations

J_z = (1 0 0, 0 0 0, 0 0 -1) <--[3x3] matrix

The Attempt at a Solution

I don't quite understand how to do this because up till now we have been trying to find expectation values for observables in same spin states.

Anyways here's my attempt:

ψ* J_z ψ = <J_z>
= ψ* (1 0 0, 0 0 0, 0 0 -1) ψ

the reason I get no further than this is because I don't see how I'm supposed to find a spin-1/2 state which would be in the form of a [2x1] matrix and then multiply the [3x3] matrix by a [2x1] matrix, what's the deal here? I'm guessing I've completely misinterpreted something but I can't see what. .

 

[cragar]

can you be more precise with your question.
going from spin-1 to spin-1/2 doesn't really make sense, can you give us more context
on what your trying to do.

 

[Lengalicious]

The exact question is as follows:
Produce a state of spin 1/2 particle such that when measuring <J_z> = 0, normalise that state. Can it be that at the same time <J_z> = 0 and <J_x> = 0?

It doesn't explicitly tell me that <J_z> is for a spin-one particle but I've just assumed that the J matrices are since that's what our lecturer has used as notation for spin-one particles and S for spin 1/2 particles. Hope this makes the question clearer?