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Matrix representation

  1. Sep 22, 2007 #1
    let X be an operator, we can write X as a matrix where:

    [tex] X = \sum_{a''} \sum_{a'} |a''><a''|X|a'><a'| [/tex]

    where [tex] <a''|X|a'> [/tex] a'' are the rows and a' are the columns. I was wondering what happened to the |a''> <a'|?

    It seems like they are disregarded when transforming to matrix notation. I was wondering why that is?
     
    Last edited: Sep 22, 2007
  2. jcsd
  3. Sep 23, 2007 #2

    malawi_glenn

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    Homework Helper

    this is "only" a representation.

    <a'| is a ket and "runs" like a column matrix; so the a' in the matrix element <a'' | X | a'> becomes the column indicies. |a''> is a bra and "runs" like a row matrix; so the a'' becomes row indices.
     
  4. Sep 23, 2007 #3

    mjsd

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    let's look at a simple example where
    [tex]|a''\rangle, |a'\rangle \in \left\{ \begin{bmatrix}1\\0 \end{bmatrix},
    \begin{bmatrix}0\\1\end{bmatrix} \right\}[/tex]
    are orthonormal sets
    so eg.
    [tex]\sum_{a'} |a'\rangle\langle a'| =
    \begin{bmatrix}1 \\ 0 \end{bmatrix}
    \begin{bmatrix}1&0 \end{bmatrix}+
    \begin{bmatrix}0 \\ 1 \end{bmatrix}
    \begin{bmatrix}0&1 \end{bmatrix} = \begin{bmatrix}1 & 0\\ 0& 1 \end{bmatrix}[/tex]
    and now

    [tex]\sum_{a', a''} |a''\rangle\langle a''| X |a'\rangle\langle a'| =
    \sum_{a', a''} \langle a''| X |a'\rangle |a''\rangle\langle a'|[/tex]
    because [tex]\langle a''| X |a'\rangle[/tex] is just a complex number at the appropriate position defined by the "index" of a' and a''
    and because
    [tex]\sum_{a', a''} |a''\rangle\langle a'| =
    \begin{bmatrix}1 \\ 0 \end{bmatrix}
    \begin{bmatrix}1&0 \end{bmatrix}+
    \begin{bmatrix}1 \\ 0 \end{bmatrix}
    \begin{bmatrix}0&1 \end{bmatrix}+
    \begin{bmatrix}0 \\ 1 \end{bmatrix}
    \begin{bmatrix}0&1 \end{bmatrix} +
    \begin{bmatrix}0 \\ 1 \end{bmatrix}
    \begin{bmatrix}1&0 \end{bmatrix}
    = \begin{bmatrix}1 & 0\\ 0& 0 \end{bmatrix} +
    \begin{bmatrix}0 & 1\\ 0& 0 \end{bmatrix}+
    \begin{bmatrix}0 & 0\\ 0& 1 \end{bmatrix}+
    \begin{bmatrix}0 & 0\\ 1& 0 \end{bmatrix}[/tex]

    the above four 2x2 matrices form a set of basis states for any generic 2x2 operator X with complex entries. As you can see from the original sum of X, for each a' and a'', the basis matrix is multiplied by the corresponding complex number defined by [tex]\langle a''| X |a'\rangle[/tex]. does this answer your question:
    all you have done in going to the index notation is implicitly assumed a set of basis matrices so that [tex]\langle i| X |j\rangle[/tex] means ij element of the matrix.
     
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