I'm confused in finding the condition for minimum uncertainty, The author in the book I refer goes on saying that
$|g\rangle=c|f\rangle$
is the condition for minimum uncertainity
for some constant $c$
Where $|f\rangle=(\hat{A}-\langle A\rangle)\psi$
and $|g\rangle=(\hat{B}-\langle B\rangle)\psi$
where $A$ and $B$ are any observables
This is acceptable due to the Schwarz inequaltity (since it becomes an equality when the angle between 2 vectors are zero), the author further says that this equality results if
${\operatorname{Re}\langle f|g\rangle}={0}$
I dont understand why this should be the condition , so if according to the author the angle should be zero, shouldn't
${\operatorname{Im}\langle f|g\rangle}$ be equal to ${0}$ ?
since for any varialbles $a$ and $b$ the argument of $z=a+ib$ can be zero only if the
$\tan^{-1}\frac{b}{a}=0$ which implies $b=0$ or $\operatorname{Im}{z}=0$
Kindly help me understand, any help is appreciated