After completing an introduction to quantum mechanics course, i am concerned that i do not fully understand what a 'fuzzy' observable is in quantum mechanics, although it has been mentioned significantly in my course. could someone explain what this means in as much detail as possible? i am realizing that while i have a strong grasp of the mathematics, i sometimes overlook the basic physical meaning on things which can lead to me arriving at correct answers, but still not completely understanding what i just did. fuzzy observables seem to be another one of these situations.

their crystal patterns have been disrupted by stress

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.