I know that an inner product between two vectors is defined like:
$$\langle a | b\rangle = {a_1}^\dagger b_1+{a_2}^\dagger b_2+\dots$$
but because a transpose of a component for example $a_1$ is again only $a_1$ the above simplifies to:
$$\langle a | b\rangle = \overline{a_1} b_1+\overline{a_2} b_2+\dots$$
Where $\overline{a_1}$ is a complex conjugate of $a_1$. Furthermore we can similarly define an inner product for two complex functions like this:
$$\langle f | g \rangle = \int\limits_{-\infty}^\infty \overline{f} g\, dx$$
In the Griffith's book (page 96) there is an equation which describes expectation value and we can write this as an inner product of a function $\Psi$ with a $\widehat{x} \Psi$:
\begin{align*} \langle x \rangle = \int\limits_{-\infty}^{\infty}\Psi\,\,\widehat{x}\Psi\,\,dx = \int\limits_{-\infty}^{\infty} \Psi\,\,(\widehat{x}\Psi)\,\, dx \equiv \underbrace{\langle\Psi |\widehat{x} \Psi \rangle}_{\rlap{\text{expressed as an inner product}}} \end{align*}
In Zettili's book (page 173) the expectation value is defined like a fraction:
\begin{align*} \langle \widehat{x} \rangle = \frac{\langle\Psi | \widehat{x} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \end{align*}
Main question: I know the meanning of the definition in Griffith's book but i simply have no clue what Zetilli is talking about. What does this fraction mean and how is it connected to the definition in the Griffith's book.
Sub question: I noticed that in Zetilli's book they write expectation value like $\langle \widehat{x}\rangle$ while Griffith does it like this $\langle x \rangle$. Who is right and who is wrong? Does it matter? I think Griffith is right, but please express your oppinion.
