# Probability of Spin in Y-up direction

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1. Apr 6, 2015

### Raynor49

1. The problem statement, all variables and given/known data
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. {Sx, Sy, Sz}, 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.

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

3. 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!

2. Apr 8, 2015

### Staff: Mentor

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?