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What is <S> ?

  1. Dec 8, 2013 #1
    I am reading about spin, and I think <S[itex]_{x}[/itex]> represents the measured value of spin along the x component. The same would follow for <S[itex]_{y}[/itex]> and <S[itex]_{z}[/itex]>. I also think I read that we can't measure all three values during an experiment. Once we measure one component, we can't measure the other two. So then I don't really understand what <S> means because it cannot be the total measured spin, since we can only know one component.

    Does <S> mean the expectation value of each component?
  2. jcsd
  3. Dec 8, 2013 #2


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    Gold Member

    [itex]<S> [/itex] is the expectation value for the total spin.The point that it can't be known exactly doesn't mean it is not defined at all.
  4. Dec 8, 2013 #3


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    Staff: Mentor

    No, it's the expectation value of the magnitude of the total spin vector, that is, the expectation value of ##\sqrt{S_x^2+S_y^2+S_z^2}##. It is somewhat counterintuitive, but we can know the magnitude of the vector as well as the value of any one of its three components.
  5. Dec 8, 2013 #4
    :smile: I'm glad I asked... thanks Shyan and Nugatory!
  6. Dec 8, 2013 #5


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    Staff: Mentor

    As noted above, it's the expectation value of the magnitude of the spin (intrinsic angular momentum) vector. For a fundamental particle this is in fact fixed for each particle: ##\langle S \rangle = S = \sqrt{s(s+1)} \hbar## where ##s## is the spin quantum number for that particle.

    For an electron, ##s = 1/2## so ##\langle S \rangle = S = (\sqrt{3}/2) \hbar##.
  7. Dec 8, 2013 #6
    jtbell, thanks for posting... that is exactly what I have just been putting together, and your post confirms it. I appreciate it!
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