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Homework Help: QM: Sum of projection operators = identity operator?

  1. Jun 19, 2010 #1

    Simfish

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    1. The problem statement, all variables and given/known data

    So we have an observable K = [tex] \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix}[/tex]

    and its eigenvectors are v1 = (-i, 1)T and v2 = (i, 1)T corresponding to eigenvalues 1 and -1, respectively.

    Now if we take the outer products, we get these...

    |1><1| = (-i, 1)T*(i, 1) = [tex] \begin{bmatrix} 1 & -i \\ i & 1 \end{bmatrix}[/tex]

    |-1><-1| = (i, 1)T*(-i, 1) = [tex] \begin{bmatrix} 1 & i \\ -i & 1 \end{bmatrix}[/tex]

    Then we add them and they sum up to form 2*(identity operator).

    But isn't it supposed to sum up to the identity operator? What's wrong, or what happened?
     
    Last edited: Jun 19, 2010
  2. jcsd
  3. Jun 19, 2010 #2

    tiny-tim

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    Hi Simfish! :wink:
    Don't you need to normalise them (to unit vectors), by multiplying by 1/√2 ? :smile:
     
  4. Jun 19, 2010 #3

    Simfish

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    Hi. :) Okay, good idea. Thanks!
     
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