Particle in an infinite potential well, showing the uncertainty in x

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Discussion Overview

The discussion revolves around a physics problem involving a particle confined in an infinite potential well, specifically focusing on deriving the uncertainty in the particle's position based on its momentum and energy levels. The scope includes theoretical reasoning and mathematical derivations related to quantum mechanics.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents the energy level formula for a particle in an infinite potential well and seeks to demonstrate the relationship between uncertainty in position and the width of the well.
  • Another participant suggests calculating momentum and applying the uncertainty principle, hinting at the use of the Hamiltonian.
  • A participant expresses confusion about the Hamiltonian but attempts to substitute it into the energy equation to find momentum.
  • Further contributions reiterate the need to apply the uncertainty principle, leading to a derived inequality involving the uncertainty in position.
  • A participant questions their calculations regarding the derived inequality and expresses uncertainty about their approach.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are varying levels of understanding and confusion regarding the application of the uncertainty principle and the calculations involved.

Contextual Notes

Some participants express uncertainty about specific terms and concepts, such as the Hamiltonian, which may affect their ability to follow the discussion. There are also unresolved mathematical steps in deriving the relationship between uncertainty in position and the width of the well.

Who May Find This Useful

This discussion may be useful for students studying quantum mechanics, particularly those grappling with concepts of uncertainty and potential wells.

SuperNomad
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I've got this question and I'm absolutely clueless, any help will be greatly appreciated:

The nth energy level for a particle of mass m confined in an infinite potential well is :

E = h^2n^2/8ml^2

where L is the width of the well and h is Planck’s constant. Assuming that the uncertainty in the particle’s momentum is equal to the momentum itself, show that the uncertainty in the particle’s position is less than the width of the well by a factor of n.
 
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Figure out it's momentum. Calculate uncertainty. Use uncertainty principle. Hint: the Hamiltonian is [tex]\frac{p^2}{2m}[/tex].
 
Don't really know what 'hamiltonian' means but anyway I substituted in p^2/2m for E

so I get p^2/2m = h^2n^2/8ml^2

re-arranging for p I get: p = hn/2l

not entirely sure where to go from here.
 
"assuming that the uncertainty in the particle’s momentum is equal to the momentum itself"
Now use the uncertainty principle.
 
so delta(x) p > h-bar/2

delta(x) hn/2L > h-bar/2

delta(x) hn/2L > h/4pi

delta(x) > L/2pi(n)

...

What have I done wrong?
 
Last edited:

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