Understanding |\beta><\alpha| in Sakurai's Book

[rasko]

In Sakurai's book, page 22:

|\beta&gt;&lt;\alpha| \doteq<br /> \left( \begin{array}{ccc}<br /> &lt;a^{(1)}|\beta&gt;&lt;a^{(1)}|\alpha&gt;^{*} &amp; &lt;a^{(1)}|\beta&gt;&lt;a^{(2)}|\alpha&gt;^{*} &amp; \ldots \\<br /> &lt;a^{(2)}|\beta&gt;&lt;a^{(1)}|\alpha&gt;^{*} &amp; &lt;a^{(2)}|\beta&gt;&lt;a^{(2)}|\alpha&gt;^{*} &amp; \ldots \\<br /> \vdots &amp; \vdots &amp; \ddots <br /> \end{array} \right)

How can people get it? Following is my idea:

|\beta&gt;&lt;\alpha|\\= |\beta&gt; (\sum_{a&#039;}|a&#039;&gt;&lt;a&#039;|)&lt;\alpha|\\<br /> =\sum_{a&#039;}(&lt;a&#039;|\beta&gt;)(&lt;\alpha|a&#039;&gt;) [STEP *]

then we get
\doteq(&lt;a^{(1)}|\alpha&gt;^{*}, &lt;a^{(2)}|\alpha&gt;^{*} ,\ldots)\cdot <br /> \left( \begin{array}{c}<br /> &lt;a^{(1)}|\beta&gt;\\<br /> &lt;a^{(2)}|\beta&gt;\\<br /> \vdots<br /> \end{array} \right)

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

 

[Gokul43201]

STEP* is incorrect. The LHS is a matrix (operator) while the RHS is a number (a scalar). What you have actually calculated (your error is in not being careful with the order) is the inner product\langle \alpha | \beta \rangle = \sum_{a&#039;} \langle \alpha | a&#039; \rangle \langle a&#039; | \beta \rangle.

For the outer product, you are (post)multiplying a row vector with a column vector (in that order). Reversing the order gives the inner product, a scalar.

 

[rasko]

Thank you!