Let $P$ be the momentum operator. Susskind writes:
$$P |\Psi \rangle =-ih \frac d {dx} |\Psi \rangle$$ Then he states that this can be rewritten as $$-ih\frac {d \psi(x)}{dx}$$ Where $\psi(x)$ is the wave function belonging to $|\Psi\rangle$
It seems to me that the first equation is meaningless. How can we differentiate a state (a ket) w.r.t. $x$? (as opposed to the representation of that state in a wave function, as in the second equation).
Am I correct that this notation $\frac d {dx} |\Psi \rangle$ is hacky?
If yes, how do we correctly write down the definition of the momentum operator in terms of the ket $|\Psi \rangle$? (i.e. without choosing a basis, and thus without using a wave function)
