Really quick question: It feels somehow wrong to write it out, so just correct me:$\\[12pt]$

$$\mathrm{<x><p>\: = \:<\psi|x|\psi><\psi|p|\psi> \:= \: <\psi|x\,p|\psi> \: = \: <x\,p>}\\[12pt]$$

The critical point probably consist of taking use of $\quad |\psi><\psi| = 1\quad$ which might not be valid.

Really quick question: It feels somehow wrong to write it out, so just correct me: $$\langle x\rangle\langle p\rangle=\langle\psi|x|\psi\rangle\langle\psi|p|\psi\rangle =\langle\psi|xp|\psi\rangle=\langle xp\rangle.$$ The critical point probably consist of taking use of $|\psi\rangle\langle \psi| = 1$, which might not be valid.