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Matrix for a Stern-Gerlach apparatus

  1. Sep 7, 2014 #1
    I am trying to find the matrix representing a modified Stern-Gerlach apparatus (as proposed in the Feynman lectures) with its magnetic field in the z direction and a filter that blocks spin 1/2 atoms that are in the |-z> state (thus I'll call the apparatus a SG+z apparatus). I want to use |+x> and |-x> as base states.

    For an atom in the |+y> = 0,707|+x> + 0,707|-x> state entering the SG+z apparatus to exit in the |+z> = 0,707|+x> + 0,707 i |-x> state, the apparatus would have to correspond to the matrix $$\left( \begin{array}{cc} 1 & 0 \\ i & 0 \end{array} \right)$$.

    The same matrix will transform the states |-y> = 0,707|+x> - 0,707|-x>, |+z>, |-z> = 0,707|+x> - 0,707 i |-x> and |+x> into the state |+z> as required, but it doesn't work for the state |-x> (it gives 0!). Am I trying to do something that is impossible or incorrect here?

    I am aware my question might need some clarification, which I will be happy to provide.
     
    Last edited: Sep 7, 2014
  2. jcsd
  3. Sep 8, 2014 #2

    kith

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    If your apparatus transmits +z and blocks -z the matrix is determined by A|+z> = |+z> and A|-z> = 0. So in the z-basis it reads
    $$\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)$$
    Note that an initially normalized superposition state won't be normalized anymore after applying the matrix. This is because the coefficients represent the probability to get the corresponding final state.
     
  4. Sep 8, 2014 #3

    atyy

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    I don't think you can get a matrix which is independent of the initial state and gets you a normalized final state, since you are describing wave function collapse, which is non-unitary. The matrix should project onto |+z>, so it should represent the operator |+z><+z| = |+x><+x| - i|+x><-x| + i|+x><-x| + |-x><-x|, which will have a matrix representation in the x-basis $$\left( \begin{array}{cc} 1 & -i \\ i & 1 \end{array} \right)$$
     
  5. Sep 8, 2014 #4
    Thank you! It made me realize I had made a mistake in my post when I wrote that the SG+z apparatus could transform state |-z> into state |+z>. The matrix you propose atyy gives the null state when applied to state |-z>, as required.
     
  6. Sep 8, 2014 #5

    atyy

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    I don't know whether my matrix is exactly right, but I'm sure kith's is (because the answer is obvious in the z-basis), so one way to check it is to check that his answer and my proposal are related by a coordinate transformation from the z-basis to the x-basis.
     
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