Probability of Spin in Y-up direction

[Raynor49]

Homework Statement

a) If a spin-1/2 particle is in the up spin state along z, what is the probability that if its spin along the y-direction is measured it will be found to be pointing in the “up” direction along y?

(b) Calculate the expectation values of the components of S, i.e. {S_x, S_y, S_z}, for a spin-1/2 particle in the state χ₊^z , i.e. spin-up along z

(c) Ditto in the state χ_-^z, i.e. spin-down along z.

Homework Equations

1) ψ ( θ) = cos(θ/2)ZUP + sin(θ/2)ZDOWN.
2) YUP = (ZUP + iZDOWN)/√2
3) YDOWN = (ZUP − iZDOWN)/√2

The Attempt at a Solution

For a), can you rewrite equations 2 and 3 like ZUP = (YUP + iYDOWN)/√2 & ZDOWN = (YUP − iYDOWN)/√2 to plug into equation 1? Since I know from equation 1 the probabilities of zup or zdown are (cos(θ/2))^2 & (sin(θ/2))^2 respectively, if you subbed in the y values for zup and zdown, could you then simplify and square the coefficients of yup and ydown to get their probabilities?

Any help is appreciated! Thanks!

 

[DrClaude]

Your approach is correct, but I don't see what you're going to do with equation 1. Can't you simplify the problem further?