Matrix Representation: What Happens to |a''><a'|?

[indigojoker]

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?

 

[malawi_glenn]

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.

 

[mjsd]

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.