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Probability of Spin in Y-up direction

  1. Apr 6, 2015 #1
    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. jcsd
  3. Apr 8, 2015 #2


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    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?
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