Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Second Measurement of Spin

  1. Sep 25, 2012 #1
    So in my quantum class we learned that if you measure spin in one direction and get h/2 and then in another direction that it will be (plus or minus)h/2 as well. I was wondering how you would know the probability of it being the positive value vs the negative value. It's a function of the angle between the two directions, right?
  2. jcsd
  3. Sep 25, 2012 #2
    I'm just a beginner myself, so wait for other replies. I've found that if spin is measured along a given axis of a spin½ particle, then the probability (p) that spin then measured along another axis will have the same sign (+ or -) is:

    p = cos(α)/2 + 50%,

    where α is the angle the new axis makes with the original axis.

    Good luck!
    Last edited: Sep 25, 2012
  4. Sep 25, 2012 #3
    That looks like it works. Do you know where this comes from? I figured it involves the pauli matrices in each of two orthogonal directions acting on the possible wavefunctions, but I wasn't sure how to arrange that into an equation.
  5. Sep 25, 2012 #4
    you wanna look at stern-gerlach experiment i think. should be some good reading.
  6. Sep 25, 2012 #5


    User Avatar
    Science Advisor

    Under a finite rotation, angular momentum states transform into each other with the aid of a unitary matrix. Naturally this matrix is called the rotation matrix, and written d(j)mm'(β) = (j m'|exp(iβ/ħ Jy)|j m) where β is the angle. For j = 1/2, d(j)mm'(β) = [tex]\left(\begin{array}{cc}cos(β/2) &sin(β/2)\\- sin(β/2)&cos(β/2)\end{array}\right)[/tex] Thus, if you have a state with Jz = +1/2 and describe it in terms of the spin states quantized along an axis inclined at an angle β to the z axis, the probability amplitude of finding Jβ = +1/2 is cos(β/2) and the probability amplitude of finding Jβ = -1/2 is sin(β/2).
  7. Sep 25, 2012 #6
    I'm afraid I made mine up from a table of some spin correlations.
  8. Sep 25, 2012 #7


    User Avatar
    Science Advisor
    Gold Member

    I think that:

    Cos^2(theta/2) = Cos(theta)/2 + .5

    Where theta = {0, 45, 60, 90 degrees} that looks pretty good. :smile:
  9. Sep 27, 2012 #8
    Interesting! My daughter's old precalculus book lists the "Double Angle Formulas":

    cos = 2cos2θ -1 = 1 - 2sin2θ = cos2θ - sin2θ, also

    sin = 2sinθcosθ

    This raised a follow-up question, if I may. (I'll start a new thread if needed.)
    Over what range of angles is the quantum spin correlation formula considered complete?

    The table I had listed five spin correlations from 0° to 180°. Classical trigonometry addresses 360° in its unit circle. But it seems that quantum spin is associated with a 720° rotation, at least for fermions.
    Last edited: Sep 27, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook