Why the uncertainty principle can be used for estimation?

Question

It is usually said/done in textbooks and classes that if $\Delta x$ is known then $\Delta p_x$ can be estimated using the uncertainty principle as $\Delta p_x \sim \hbar/\Delta x$.

But the uncertainty principle does not say that, it says $\Delta p_x\ge\hbar/\Delta x$. That means we can only set a lower bound on $\Delta p_x$, i.e. $\Delta p_x$ could be anything between $\hbar/\Delta x$ and $\infty$

Why then the lower bound is chosen for estimation?

Are there certain situations where the products in uncertainties is of the same order as $\hbar$?

Answer 1

It assumes a sort of democracy between $x$ and $p$, and is obviously not valid everywhere. The connection between classical and quantum mechanics happens through coherent states, which are states which minimize the product $\Delta p \Delta x$, and in a sense, behave most classically (and have well-defined classical limits as $\hbar \to 0$).

So if you started with a something behaving reasonably classically, e.g. an electron's motion in a uniform electric field, and you wanted to estimate the uncertainty of it's momentum, you could use this as a starting point. There's nothing really rigorous about it, other than the fact that coherent states seem to enjoy a privileged status in nature.