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Homework Help: How to find this spin component

  1. Sep 16, 2007 #1
    We know that [tex] S_x = \frac{\hbar}{2} \left( |+ \rangle \langle - | + | - \rangle \langle+| \right)[/tex]

    But what is [tex]|S_x ; + \rangle[/tex]?

    I think my text says [tex]|S_x ; + \rangle = \frac{1}{\sqrt{2}} \left( |+ \rangle + | - \rangle \right)[/tex] but i dont know how they got this.

    I feel like this is a trivial question but I'm not sure how one finds [tex]|S_x ; + \rangle[/tex]
     
  2. jcsd
  3. Sep 17, 2007 #2

    dextercioby

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    But WHAT is [itex] |S_{x},+\rangle [/itex] ?? I've never seen this notation before...And it's not that i've looked into one book...:rolleyes: I haven't looked in your book, apparently, you might share with us the title and the author...
     
  4. Sep 17, 2007 #3
    I imagine that it's the positive spin direction for S_x.

    OP: it's just an eigenvector -- so you find it in the same way that you find any eigenvectors. If it helps, write S_x as a matrix, in the |+>, |-> basis that you've got things in.
     
  5. Sep 17, 2007 #4
    I just figured it out.

    I am using * as the dot product

    [tex]S * \hat n | S * \hat n ; + \rangle = \frac{ \hbar}{2} | S * \hat n ; + \rangle [/tex]
    [tex]| S * \hat n ; + \rangle = \cos \frac{\beta}{2} |+ \rangle + \sin \frac{\beta}{2} e^{i \alpha} | - \rangle [/tex]

    where beta is the polar angle and alpha is the azimuthal angle.

    therefore, an S_x measurement would be where beta = pi/2 and alpha =0

    since the S_x measurement would yield +hbar/2, we get:

    [tex]| S_x; + \rangle = \cos \frac{\pi/2}{2} |+ \rangle + \sin \frac{\pi/2}{2} e^{0} | - \rangle [/tex]

    therefore:
    [tex]|S_x ; + \rangle = \frac{1}{\sqrt{2}} \left( |+ \rangle + | - \rangle \right)[/tex]
     
    Last edited: Sep 17, 2007
  6. Sep 17, 2007 #5

    nrqed

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    They want the state which is an eigenstate of Sx with the eigenvalue +hbar/2.

    So you could write [tex]|S_x ; + \rangle = \alpha
    |+ \rangle + \beta |-
    \rangle [/tex]

    and apply S_x, imposing [itex] S_x |S_x; + > = \frac{\hbar}{2} |S_x;+> [/itex] and then solve for alpha and beta (and normalize at the end)
     
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