Uncertainty in the position of a particle, and momentum question

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SUMMARY

The discussion centers on the relationship between the uncertainty in the position of a particle and its momentum, specifically when the uncertainty in position is comparable to the de Broglie wavelength. Participants confirm that the uncertainty in momentum, denoted as Δp, is proportional to the particle's momentum when Δx is similar to the wavelength. The key equations referenced include Δx ∼ λ and Δx Δp ∼ ℏ, where ℏ is the reduced Planck's constant. This establishes a clear connection between quantum mechanics and the behavior of particles at a fundamental level.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly the Heisenberg uncertainty principle.
  • Familiarity with de Broglie wavelength and its significance in particle physics.
  • Basic knowledge of momentum and its relationship to wavelength in quantum contexts.
  • Proficiency in mathematical manipulation of physical equations, including the use of Planck's constant.
NEXT STEPS
  • Study the Heisenberg uncertainty principle in detail, focusing on its implications in quantum mechanics.
  • Explore the derivation and applications of the de Broglie wavelength in particle physics.
  • Learn about the reduced Planck's constant (ℏ) and its role in quantum mechanics.
  • Investigate momentum-energy relationships in quantum systems, particularly in wave-particle duality.
USEFUL FOR

Students in physics, particularly those studying quantum mechanics, as well as educators and professionals seeking to deepen their understanding of particle behavior and uncertainty principles.

Zit501
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Hi all,

I just can't wrap my head around this one. It should be fairly easy, but I'm lost!

Homework Statement


Suppose that the uncertainty in the position of a particle is of the order of its de Broglie wavelength. Show that in this case the uncertainty in its momentum is of the order of it's momentum.


Homework Equations



Not 100% sure...

Delta x is similar to the wavelength
Delta p is similar to Plancks constant over wavelength
Therefore delta x delta p are similar to Plancks constant

The Attempt at a Solution


Well I can see how delta x and delta p can be related together to the wavelength, but after that I'm lost.

Any help much appreciated! I'm a B.Sc Mechanical Engineer doing an entry level nuclear physics course and I'm holding on... just.
 
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Yes, so you almost got it.

[tex]\Delta x \sim \lambda[/tex]
If the uncertainty relation is near the minimum value, then
[tex]\Delta x \Delta p \sim \hbar[/tex]

So what is [itex]\Delta p[/itex] then. What is [itex]\lambda[/itex] in terms of p?
 

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