No, quantum mechanics does not state that you cannot simultaneously measure both position and momentum precisely. It is a consequence of the theory, but it is not what the theory is based on.

Quantum mechanics states that a classical position, classical momentum, or other classical observables do not exist except in the rare cases when the quantum object interacts with something classical (such as an instrument.)

When you look at the mathematics (and you have to look at the mathematics; quantum mechanics cannot be intuited) something amazing emerges. The formal equations of quantum mechanics, such as the Schrödinger equation, can be “derived” easily from classical physics. However, this equation offers many more solutions than its classical counterpart. Quantum mechanics begins when we look at these solutions and accept them as valid descriptions of reality, despite the fact that they seemingly make no intuitive sense, certainly not in the context of classical physics.

Now you may wonder, what on Earth possesses us to go down this rabbit hole? Very simple: physics is based on experiment and observation. And we found that this is how the physical world works.

When we look at this much richer world of quantum solutions, we find that indeed, most of the time that particle does not have a classical position or a classical momentum. Moreover, the math tells us, when it is confined to a classical position by a measurement, its classical momentum does not exist; it remains in a superposition of states.

So when you think of an electron inside a cathode ray tube, going from the cathode to the screen while mysteriously going through two holes at the same time, and ask yourself, “What was the electron’s path?”, unfortunately the only legitimate answer sounds just as mysterious as the little boy telling Neo in the film The Matrix that there is no spoon: There is no (classical) path. It’s not that we cannot measure it. It truly does not exist. And whether we like it or not, that’s the way Nature works. But there is one advantage that we have over a piece of fiction like The Matrix: our outlandish statement is grounded in firm mathematics that leads to testable predictions, through which our outlandish claims can be  (and have been, countless times) verified and validated.