What is uncertainty principle exactly?

Question

I've learned a little about uncertainty principle. According to words on Internet, it says that the position and the velocity of an object cannot both be measured exactly at the same time. And there are some examples like:

Any attempt to measure precisely the velocity of a subatomic particle, such as an electron, will knock it about in an unpredictable way, so that a simultaneous measurement of its position has no validity.

This has been bothering me for a long time: If the uncertainty is cauesed by measuring instruments, then we can't say that particles (or everything) themselves are uncertain. So can I say the world is actually certain, and the reason why it looks uncertain is we can't predict exactly?

Answer 1

The uncertainty principle has a somewhat misleading name. If you're seeking everyday analogies, the UP has less to do with inaccuracy and is much more to do with noncommutativity: a phenomenon illustrated by the obvious truth that if you put your shoes on before your socks, you have a decidedly different outcome from when you put your socks on before your shoes.

This kind of phenomenon is inherent in the basic postulates of quantum mechanics: measurements are represented by observables which are operators that act on the quantum state space together with the following special recipe for interpreting them: a measurement is modelled by the "collapse" of the quantum state to one of the operator's eigenvectors and at the same time the measurement value returned is the eigenvalue corresponding to the corresponding eigenvector. The collapse is random and its statistics are governed by the quantum state just before the measurement: the $m^{th}$ moment of the probability distribution being $\langle \psi|\hat{A}^m|\psi\rangle$ where $\hat{A}$ is the operator in question. Because operators ("matrices") on Hilbert spaces do not commute, if we impart two measurements with observables $\hat{A}$ and $\hat{B}$ one after the other, the statistics of the two measurements depend on the order the measurements are made. The outcome of simultaneous measurements in such a sequence is independent of the order of measurement, and thus there is no uncertainty in the latter measurement after the first has been made, if and only if the corresponding observables commute.