Particle in a Box: Solving Qs on Lowest Energy & Highest Probability

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Discussion Overview

The discussion revolves around solving two questions related to a particle in a one-dimensional infinite square well, focusing on energy levels and probability distributions. The scope includes theoretical aspects of quantum mechanics and mathematical reasoning related to wave functions and energy calculations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to determine the lowest energy level for a particle in a box of width (1/4)L compared to a particle in a box of width L, seeking an expression in terms of Eo.
  • Another participant notes that the energy levels for a particle in an infinite square well can be expressed as E_n = n^2 (π²ħ²)/(2mL²) for n = 1, 2, 3, ...
  • A participant suggests writing down the probability distribution for a particle in a box for an arbitrary state n and substituting n = 11 to find the first value of x where the probability is highest.
  • One participant provides the energy expression for the smaller width box, indicating that E_n = 4n² (π²ħ²)/(2mL²) = 4n² E₀, and encourages others to deduce further from this point.

Areas of Agreement / Disagreement

Participants present various approaches to the problems, but there is no consensus on the solutions or methods to be used. Multiple viewpoints and methods are discussed without resolution.

Contextual Notes

Participants rely on specific definitions and mathematical expressions related to quantum mechanics, which may not be universally agreed upon. The discussion includes assumptions about the infinite square well model and the implications of wave functions on probability distributions.

Physicsiscool
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Can someone please explain to me how I should go about solving the following 2 questions?

#1 The lowest energy level in a particle confined to a one-dimensional region of space with fixed dimension "L" is Eo. If an identical particle is confined to a similar region with the fixed distance (1/4)L, what is the energy of the lowest energy level that the particles have in common? Express in terms of Eo.

#2 Consider a particle in a box of width "L" and let the particle be in a state n = 11. What is the first value of x, larger then 0, where the probability of finding the particle is the highest?
 
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If these are questions concerning the infinite square well, use the fact that the energy levels are:
[tex]E_n=n^2<br /> \frac{\pi^2\hbar^2}{2mL^2}[/tex]
for n=1,2,3,...
 
Physicsiscool said:

#2 Consider a particle in a box of width "L" and let the particle be in a state n = 11. What is the first value of x, larger then 0, where the probability of finding the particle is the highest?


Write down the probability distribution for the general case with arbitrary n. (You may need to start with the wave function and find the probability distribution from that.) Substitute n = 11. For the particle in a box, it should be pretty obvious where the maxima are, from the general form of the probability distribution. Sketching a graph of it might help.
 
FOr the first part of the problem

[tex]E_{0} = \frac{\pi^2\hbar^2}{2mL^2}[/tex]

for the smaller width box

[tex]E_{n} = 4n^{2}\frac{\pi^2\hbar^2}{2mL^2} = 4n^2 E_{0}[/tex]

from here figure it out.
 

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