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I was reading about Integral equations, and I found this excerpt in Portuguese Wikipedia:

"integral equations serve to determine the position in all instances of an object, if known, its instantaneous velocity at all times"

But reading about the Uncertainty Principle, I found this excerpt:

For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

What's wrong in my conclusion? Really the two principles contradict each other?

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One is classical mechanics, the other is quantum. That's it. –  jinawee May 16 at 13:31

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If you know its instantaneous velocity at every moment of time in a given interval AND you knew the exact position of a particle at the start of the interval, then you could use integrals to find its exact position at any moment in the interval. That is a logically true statement. There is no contradiction here with the Uncertainty principle because you cannot know the exact position and the instantaneous velocity at the start of the interval. So while the conditional statement is logically true, the uncertainty principle makes the antecedent false, which gives no information about the consequence and allows there to be no contradiction.

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It's really. Thanks. @jinawee said something that is also correct: "One is classical mechanics, the other is quantum." –  Only a Curious Mind May 16 at 14:48
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@LucasAbilidebob there are integral forms of the quantum equations. They start with a cloud of probability at $t=0$, and give you the evolution of that fuzzy cloud along time. –  Davidmh May 16 at 18:24
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@Lucas Integrals are a mathematical tool, they're not inherently classical or quantum, and in fact are used extensively in both theories. So it's more like, in classical mechanics, integrals allow you to determine the position of a particle over time given its velocity-graph. They cannot be used in QM for that same purpose (due to the uncertainty principle), but they are used for many other calculations. –  BlueRaja - Danny Pflughoeft May 16 at 19:36

MASS is the answer to your mystery. If you write out the equations you notice that The Heisenberg equation refers to MOMENTUM, which is the product of mass and velocity ( a vector - magnitude and direction). Whereas, Newtonian integrals are generally integrated with respect to a scalar value such as VELOCITY, or time.

The other issue with the comparison is that you are meshing quantum physics with Newtonian physics, which can differ in areas. With Heisenberg you have to keep in mind that the objects of interest are super super tiny, low mass sub-atomic particles whereas Newtonian physics applies to marbles and airplanes. In order=r to "view" a subatomic particle you have to shine some sort of light at it. Light particles called photons carry a super super tiny mass as well. So, when you "view" a subatomic particle the light needed view it results in collisions between two masses, which can affect momentum....or a predicted position.

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