Uncertainty Principle of a high speed particle

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Discussion Overview

The discussion centers around the implications of Heisenberg's Uncertainty Principle in relation to high-speed particles, specifically exploring the connection between a particle's speed and the uncertainty of its velocity. Participants examine theoretical aspects and mathematical formalism related to wave mechanics and measurement uncertainty.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes that according to Heisenberg's Uncertainty Principle, increased certainty in a particle's position leads to increased uncertainty in its momentum, suggesting that this can be interpreted in terms of velocity.
  • Another participant questions whether high speeds make it more difficult to measure velocity precisely, proposing that high uncertainty results in a wide range of measured speed values.
  • A participant proposes a connection between the range of speed values and the concept of wave localization, suggesting that combining waves of different wavelengths leads to a more localized wave function but results in a spread-out momentum representation.
  • Another participant agrees with the wave mechanics explanation, indicating that transforming a position wave function with a steep peak results in a wide flat shape in momentum space, and vice versa.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and agreement on the implications of wave mechanics in relation to uncertainty, but no consensus is reached on the specifics of how high speeds affect measurement uncertainty.

Contextual Notes

The discussion involves complex concepts from quantum mechanics and wave mechanics, with participants referencing mathematical formalism without resolving all underlying assumptions or dependencies on definitions.

Misha Kuznetsov
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Hello everyone,
Heisenberg's Uncertainty Principle states that the less uncertainty there is of a particles position, the more uncertainty there has to be of its momentum. Since mass is a constant in this case, we can refer to the uncertainty of the velocity instead. I was reading a physics book, The Feynman Lectures on Physics, and it mentioned, "if we try to pin down a particle by forcing it to be at a particular place, it ends up by having a high speed."

My question is, what does high speed of a particle have to do with the uncertainty of its velocity? Is it that it is more difficult to precisely measure the velocity of a particle at high speeds?
 
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Misha Kuznetsov said:
My question is, what does high speed of a particle have to do with the uncertainty of its velocity? Is it that it is more difficult to precisely measure the velocity of a particle at high speeds?

A high uncertainty means that if you measure the speed many times (more precisely, you repeat the experiment from the beginning many times, making one measurement each time) you'll get a wide range of values. The average of these values has to be far from zero, as otherwise the range wouldn't be wide. Thus, if you measure the speed, you expect to get a value that is far from zero... and that's a high speed.
 
Okay, I think I understand now. Is there a wide range of values for the speed because adding many waves of various wavelengths yields one that is more localized, but its momentum (wavelength) becomes spread out and less constant? And so when you measure many times, you find the wavelength from different waves each time?
 
Misha Kuznetsov said:
Okay, I think I understand now. Is there a wide range of values for the speed because adding many waves of various wavelengths yields one that is more localized, but its momentum (wavelength) becomes spread out and less constant? And so when you measure many times, you find the wavelength from different waves each time?

Yes, that's pretty much how the math works out when you use the basic formalism of wave mechanics and the Schrödinger equation in the position and momentum bases - transform a position wave function with a steep peak around one location to the momentum basis and you get a wide flat shape and vice versa.
 
Thank you so much, that helped a lot. :smile:
 

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