In Sakurai's book, page 22:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]|\beta><\alpha| \doteq

\left( \begin{array}{ccc}

<a^{(1)}|\beta><a^{(1)}|\alpha>^{*} & <a^{(1)}|\beta><a^{(2)}|\alpha>^{*} & \ldots \\

<a^{(2)}|\beta><a^{(1)}|\alpha>^{*} & <a^{(2)}|\beta><a^{(2)}|\alpha>^{*} & \ldots \\

\vdots & \vdots & \ddots

\end{array} \right)[/tex]

How can people get it? Following is my idea:

[tex]|\beta><\alpha|\\= |\beta> (\sum_{a'}|a'><a'|)<\alpha|\\

=\sum_{a'}(<a'|\beta>)(<\alpha|a'>) [STEP *][/tex]

then we get

[tex]\doteq(<a^{(1)}|\alpha>^{*}, <a^{(2)}|\alpha>^{*} ,\ldots)\cdot

\left( \begin{array}{c}

<a^{(1)}|\beta>\\

<a^{(2)}|\beta>\\

\vdots

\end{array} \right)[/tex]

Is the STEP* right? I'm not sure if i have understood the ruls of ket and bra.

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# Outer product

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