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Pauli spin matrices, operating on |+> with Sx

  1. Jun 15, 2011 #1

    L-x

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    1. The problem statement, all variables and given/known data
    What is the result of operating on the state |+> with the operator Sx?

    here, |+> denotes the eigenstate of Sz with eigenvalue 1/2. I am working in units where h-bar is 1 (for simplicity, and because I don't know how to type it)

    2. Relevant equations
    [tex]S_i = \frac{1}{2} σ_i [/tex]


    3. The attempt at a solution
    My understanding of the physics of the problem is that after measurement the system will be in state where the x component of its spin is certain, as [Sx,Sz] != 0 this means it will be in some superposition of the states |+> and |-> My intuition tells me that the probability for the particle to be found in either state should be equal.

    However, operating with the matrix representation of Sx :[tex]\frac{1}{2}
    \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)
    [/tex]
    on |+> just gives (1/2)|->

    What am i doing wrong?
     
  2. jcsd
  3. Jun 15, 2011 #2

    L-x

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    I think I understand where I've gone wrong now. The problem is in my intuition, not in my appilcation of the pauli matrices.

    as [tex]|+> = \frac{1}{\sqrt{2}}(|+_x> + |-_x>) [/tex]
    (where [tex]|+_x> and |-_x>[/tex] are the eigenvectors of S_x with eigenvalues +/- 1/2)

    when we operate with Sx we obtain [tex](2^{-\frac{1}{2}})(\frac{1}{2}|+_x> - \frac{1}{2}|-_x>) [/tex] which is equal to [tex]\frac{1}{2}|->[/tex]. The confusion I had was due to a faliure to distinguish between operation with a linear operator and measurement of an ovservable.
     
    Last edited: Jun 15, 2011
  4. Jun 18, 2011 #3

    G01

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    Homework Helper
    Gold Member

    Yes, your reasoning is correct, and you have the correct result.
     
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