Please help Problems on particle-in-a-box models

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Homework Help Overview

The discussion revolves around a particle-in-a-box model in quantum mechanics, specifically focusing on calculating the probability of finding a particle within a specified region of a one-dimensional box defined by a normalized wave function. The context involves understanding the implications of the wave function and the associated probability calculations.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of the wave function to calculate the probability of locating a particle between specific bounds. Questions arise regarding the application of the integral formula for probability and the implications of using a variable length L instead of a numerical value.

Discussion Status

Participants are actively engaging with the problem, with some providing hints and others expressing confusion about the calculations. There is a request for clarification on the use of formulas and the need for detailed calculations to facilitate further assistance.

Contextual Notes

There is mention of a hint involving an indefinite integral, and participants are navigating the challenge of working with symbolic variables rather than concrete numbers. The discussion reflects uncertainty about how to proceed with the calculations given the parameters of the problem.

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The normalized wave function for a particle in a 1D box in which the potential energy is zero
between x= 0 and x= L and infinite anywhere else is

normalized wave function = sqrt(2/L)*sin(npix/L)

What is the probability that the particle will be found between x= L/4 and x= L/2 if the particle is in the state characterized by the quantum number n= 1?




Hint given:

indefinite integral of sin^2(ax)dx = .5x - .25asin(2ax)
 
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What you probably don't know (but should know) is that
P(particle between a and b)=∫_a^b |ψ(x)|^2 dx
where ψ is the wavefunction of the particle.
 
I am confused as to what I use the second formula for... How do I find the probability of the particle being located in the region bounded by x=l/4 and x=l/2? How do i get an actual value for this since L is being used, and not a real number?
 
How about you do the calculation and see what you end up with? There is a possibility that L cancels out...
Btw.: Which one is the second formula for you?
 
indefinite integral of sin^2(ax)dx = .5x - .25asin(2ax)

that is the second formula. and I tried working it out but I couldn't get L to cancel out.. i just got an ugly answer...
 
Please write down your entire calculation then I can help you with it.
 

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