Particle in an infinite potential well

In summary, the nth energy level for a particle of mass m in an infinite potential well is given by an equation involving the width of the well and Planck's constant. By assuming that the uncertainty in the particle's momentum is equal to the momentum itself, it can be shown that the uncertainty in the particle's position is less than the width of the well by a factor of n. The uncertainty principle is used to derive this result. The solution involves substitution and rearrangement, and the final answer is L/(2*pi*n), which is in agreement with the expected answer. The assumption of positive values does not affect the solution.
  • #1
mrausum
45
0

Homework Statement



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

45c2b9fd81beaf323e353898e6ee4095.png


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.

Homework Equations



Uncertainty principle?

The Attempt at a Solution



I don't really know where to start. Help!
 
Last edited:
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  • #2
I have an idea of what I'm doing now, although I've got my answer as change in x <= L/(2*pi*n) instead of L/n which seems to be what the question expects.

My method was to sub in change in P = p into the uncertainty principle, then sub that into the equation given. Then i just rearranged. What am i doing wrong?

I'm also assuming all the values are positive so the less than sign doesn't have to be flipped?
 
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  • #3
Your answer seems correct, as far as I can tell.
 
  • #4
Redbelly98 said:
Your answer seems correct, as far as I can tell.

Oh ok thanks. I guess my answers still in agreement that the uncertainties less than 1/n * L.
 

FAQ: Particle in an infinite potential well

1. What is a "Particle in an infinite potential well"?

A particle in an infinite potential well is a theoretical model used in quantum mechanics to describe the behavior of a particle confined within a potential well with infinitely high walls. It is often used as a simplified representation of a particle trapped within a small region, such as an electron in an atom.

2. How does a particle behave in an infinite potential well?

In this model, the particle can only exist within the boundaries of the potential well and cannot escape. The particle's behavior is described by a wave function, which represents the probability of finding the particle at a given position within the well. The wave function has specific boundary conditions that dictate the particle's energy levels and allowed states.

3. What are the main applications of the "Particle in an infinite potential well" model?

The "Particle in an infinite potential well" model is used to understand various quantum phenomena, such as the quantization of energy levels, tunneling, and confinement of particles. It also has practical applications in fields such as nanotechnology, where particles can be confined within small spaces to control their behavior.

4. How is the "Particle in an infinite potential well" model related to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the position and momentum of a particle cannot be known simultaneously with absolute certainty. This principle is related to the "Particle in an infinite potential well" model because the confinement of the particle within the well results in a limited range of allowed momentum values, leading to uncertainty in the particle's position.

5. What are the limitations of the "Particle in an infinite potential well" model?

The model assumes ideal conditions, such as perfect confinement and infinitely high potential walls, which may not accurately reflect real-world situations. Additionally, the model does not take into account other factors that may affect the behavior of particles, such as interactions with other particles or external forces. Therefore, it is a simplified representation and cannot fully describe all quantum systems.

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