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I was just listening to the Leonard Susskind lectures on youtube on qunatum mechanics basics. He said that uncertainty principle in classical mechanics is completely different from that of quantum mechanics. I have done some enough researches on it regarding the view but failed to understand what he supposed to mean. Almost all the articles on internet show links between uncertainty principle and quantum mechanics. I see there is nowhere written any connection between classical mechanics and uncertainty principle. If anybody knows please kindly share any lecture or preview where i can get the clear idea about the differences imposed on uncertainty principle.

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  • $\begingroup$ Link and minute of youtube video? $\endgroup$ – Qmechanic May 12 '19 at 13:42
  • $\begingroup$ youtu.be/2h1E3YJMKfA $\endgroup$ – Lonely walker May 12 '19 at 13:44
  • $\begingroup$ Are you talking about this section of the lecture, @F.sharmin? $\endgroup$ – Abhay Hegde May 12 '19 at 13:51
  • $\begingroup$ Yes sir. Glad u noticed $\endgroup$ – Lonely walker May 12 '19 at 14:12
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Uncertainty found in classical physics which he mentions in the lecture here points out the defect in our measurement procedure or devices. [Sometimes, Liouville's theorem is termed as the uncertainty principle of classical mechanics.]

Suppose we are studying the dynamics of a swarm of particles. An accurate description of initial position and momentum would be sufficient to determine the position and momentum of each classical particle at a later point of time, according to classical physics. We could apply Newton's laws to predict the movements without disturbing the system. Of course, it would be tedious to keep track of gazillion number of particles, nevertheless not impossible at least in principle. The distribution in the classical picture reflects our ignorance of the system rather than any intrinsic uncertainty.

On the other hand, the uncertainty principle of quantum mechanics is deeper and more fundamental. Stated in simple terms, it is impossible to measure simultaneously both position and momentum to arbitrary precision or accuracy.

Mathematically, it says $\Delta x \Delta p \geq \dfrac{\hbar}{2}$, where $\hbar$ is of the order of $10^{-34} \text{J s}$. It is not a limitation of our technological prowess or our 'laziness' in measuring procedure, but the nature of reality. It is not that we are incapable of measuring exact momentum and position simultaneously, but it is that we can never. Well-defined position and well-defined momentum are mutually exclusive properties for a particle at an instant of time.

In fact, this is true for not just position and momentum, but also for any pair of non-commuting operators.

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