My QM text defines the position operator as follows:
The position operator $X= (X_1,X_2,X_3)$ is such that for $j=1,2,3: \ X_j \psi(x,y,z)= x_j \psi(x,y,z)$.
To me this can mean two things.
1) $X$ is a vector and acts as $X \psi(x,y,z)= (x \psi(x,y,z), y \psi(x,y,z), z \psi(x,y,z))$. But this doesn't make sense as $X$ is an observable/operator and so must send vectors to vectors (here functions).
2)There are three position operators $X_1, X_2, X_3$ and each act as defined.
How does the postion operator act on a state? Could anyone help me out here? Thanks!