Heisenberg's Momentum-Position Uncertainty Principle

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SUMMARY

Heisenberg's Momentum-Position Uncertainty Principle establishes that it is impossible to simultaneously determine the position and momentum of a particle with unlimited precision, as articulated in Serway/Moss/Moyer. The mathematical representation of this principle is given by the inequality ΔpΔx ≥ h/2, where h represents Planck's constant divided by 2π. The discussion highlights the relationship between large wavenumbers (Δk) and the uncertainty principle, emphasizing the role of Fourier transforms in understanding this concept. The principle illustrates that precise measurement of one property inherently limits the precision of the other.

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  • Understanding of quantum mechanics fundamentals
  • Familiarity with Fourier transforms
  • Knowledge of Planck's constant and its significance
  • Basic concepts of wave-particle duality
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  • Explore the mathematical derivation of the uncertainty principle
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Koshi
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I was reading about how Heisenberg found out that it is "impossible to determine simultaneously with unlimited precision the position and momentum of a particle" (Serway/Moss/Moyer, 174)

\Delta p\Delta x \geq[STRIKE]h[/STRIKE]/2 (where [STRIKE]h[/STRIKE] is plank's constant over 2pi.)

My question is why is this true? I read that it had something to do with the large wavenumbers \Deltak, but I'm unsure exactly how that affects anything. I'm just a little hazy on the reason for why, even ignoring the error caused by measuring insturments, it would be impossible to measure two precise things at once.
 
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One way of gaining some intuition is to consider Fourier transforms. It is impossible to exactly know the frequency of a sine wave unless you have an infinitely long interval to measure it. To know its frequency (energy) you give up all localization in time. The same holds in the quantum world.
 

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