QM - Part 1 - Eigenvalues of the Hermitian Operator A are real

Hi FOlks,

Studying this theorem 'Eigenvalues of the Hermitian Operator A are real' from sakurai's Modern QM's

$\displaystyle \hat A | a' \rangle = a'| a' \rangle $

If we multiply the above by $\displaystyle \langle a''|$ from the left and right respectively we get

$\displaystyle \langle a''|\hat A|a' \rangle= \langle a''|a'|a' \rangle$ (1)and

$\displaystyle

\hat A|a' \rangle \langle a''|=a'|a' \rangle \langle a''|

$ (2)

If we flip the LHS of (1) we get

$\displaystyle

\langle a'|\hat A|a'' \rangle=\underbrace{\langle a'|(a')^*|a'' \rangle }_\textrm{?}= (a')^* \langle a'|a'' \rangle

$

Should underbrace not be $\displaystyle \langle a'|(a')^*|a'' \rangle ^*$

Second question I have is how the following was arrived at?

$\displaystyle

\langle a'| \hat A|a'' \rangle =\langle a'|a''a'' \rangle = a'' \langle a'|a'' \rangle

$

Thanks