Uncertainty Principle of a high speed particle

[Misha Kuznetsov]

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?

 

[Nugatory]

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.

 

[Misha Kuznetsov]

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?

 

[Nugatory]

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.

 

[Misha Kuznetsov]

Thank you so much, that helped a lot.