indigojoker said:
let X be an operator, we can write X as a matrix where:
X = \sum_{a''} \sum_{a'} |a''><a''|X|a'><a'|
where <a''|X|a'> 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?
let's look at a simple example where
|a''\rangle, |a'\rangle \in \left\{ \begin{bmatrix}1\\0 \end{bmatrix}, <br />
\begin{bmatrix}0\\1\end{bmatrix} \right\}
are
orthonormal sets
so eg.
\sum_{a'} |a'\rangle\langle a'| = <br />
\begin{bmatrix}1 \\ 0 \end{bmatrix}<br />
\begin{bmatrix}1&0 \end{bmatrix}+<br />
\begin{bmatrix}0 \\ 1 \end{bmatrix}<br />
\begin{bmatrix}0&1 \end{bmatrix} = \begin{bmatrix}1 & 0\\ 0& 1 \end{bmatrix}
and now
\sum_{a', a''} |a''\rangle\langle a''| X |a'\rangle\langle a'| =<br />
\sum_{a', a''} \langle a''| X |a'\rangle |a''\rangle\langle a'|
because \langle a''| X |a'\rangle is just a complex number at the appropriate position defined by the "index" of a' and a''
and because
\sum_{a', a''} |a''\rangle\langle a'| = <br />
\begin{bmatrix}1 \\ 0 \end{bmatrix}<br />
\begin{bmatrix}1&0 \end{bmatrix}+<br />
\begin{bmatrix}1 \\ 0 \end{bmatrix}<br />
\begin{bmatrix}0&1 \end{bmatrix}+<br />
\begin{bmatrix}0 \\ 1 \end{bmatrix}<br />
\begin{bmatrix}0&1 \end{bmatrix} +<br />
\begin{bmatrix}0 \\ 1 \end{bmatrix}<br />
\begin{bmatrix}1&0 \end{bmatrix} <br />
= \begin{bmatrix}1 & 0\\ 0& 0 \end{bmatrix} +<br />
\begin{bmatrix}0 & 1\\ 0& 0 \end{bmatrix}+<br />
\begin{bmatrix}0 & 0\\ 0& 1 \end{bmatrix}+<br />
\begin{bmatrix}0 & 0\\ 1& 0 \end{bmatrix}
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 \langle a''| X |a'\rangle. does this answer your question:
what happened to the |a''> <a'|?
all you have done in going to the index notation is implicitly assumed a set of basis matrices so that \langle i| X |j\rangle means ij element of the matrix.
