I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\rangle=\frac{1}{ih}\langle[A,H]\rangle$. This is famously analogous through canonical quantization of Poisson Brackets to the fact that in Hamiltonian mechanics one has $\frac{dF}{dt}=\{F,H\}$.
One has that the Hamilton equations have the following analogous form:$$\partial_t\langle x\rangle=\frac{1}{ih}\langle[x,H]\rangle=\frac{1}{m}\langle p_j\rangle$$ $$\partial_t\langle p\rangle=\frac{1}{ih}\langle[p,H]\rangle=-\langle\partial_x U\rangle$$ Whereas the first is completely analogous to the first Hamilton eq., to have the complete analogy one needs to have $$\langle\partial_x U(x)\rangle=\partial_{\langle x\rangle} U(\langle x\rangle) $$ but this happens for at most quadratic potentials. But this must mean that at the classical level $\langle x^2\rangle=\langle x\rangle\langle x\rangle$, whereas $\langle x^2\rangle\neq\langle x\rangle\langle x\rangle$ at the quantum level. In some sense then at quantum level we have a kind of correlation.
Is there anything to this correlation concept?
Indeed, at the quantum level $$ (\Delta x) ~^2= \langle (x-\langle x\rangle)^2 \rangle =\langle x^2\rangle - \langle x\rangle^2 >0 . $$
Taking, w.l.o.g., $\langle x\rangle =0$ to simplify out irrelevancies, the only state with zero width, $\langle x^2\rangle=0$ has the δ-function as a wavefunction, so then it is $|x=0\rangle$, an un-normalizable state which is not in the Hilbert space, and is unphysical (albeit useful as a component of physical ones), $$ \langle x|0\rangle = \delta (x). $$
All physical wavefunctions have non-vanishing width, typically of $O(\sqrt{\hbar})$ — which is what makes trajectories untenable in quantum mechanics. It's at the heart of the uncertainty principle, as stated. (There are easier ways to see this in phase-space quantization, to be sure.)