Originally Posted by

**manjohn12** Suppose we want to compute $\displaystyle \langle \alpha| \beta \rangle $. Let $\displaystyle |\alpha \rangle = \sum_{n} a_{n}| \phi_{n} \rangle $ and $\displaystyle |\beta \rangle = \sum_{n} b_{n}| \phi_{n} \rangle $.

So $\displaystyle \langle \alpha| \beta \rangle = \alpha^{*} \beta $. In expanding $\displaystyle |\beta \rangle $, how do we get $\displaystyle \langle \alpha| \beta \rangle = \sum_{n} b_{n} \langle \alpha| \phi_{n} \rangle $? Because by linearity this is equaled to $\displaystyle \sum_{n} \alpha \langle b_{n}| \phi_{n} \rangle $. But shouldn't is be $\displaystyle \alpha^{*} $? Because then we eventually get:

$\displaystyle \langle \alpha| \beta \rangle = \sum_{n} b_{n} \left(\sum_{m} a_{m}^{*} \langle \phi_{m}| \phi_{n} \rangle \right) = \sum_{n} a_{n}^{*} b_{n} $.