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

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In summary, the normalized wave function for a particle in a 1D box with a potential energy of zero between x=0 and x=L and infinite anywhere else is given by sqrt(2/L)*sin(npix/L). To find the probability of the particle being located in the region between x=L/4 and x=L/2, we use the formula P(particle between a and b) = ∫a^b|ψ(x)|^2 dx, where ψ is the wavefunction of the particle. There is also a hint given, stating that the indefinite integral of sin^2(ax)dx is .5x - .25asin(2ax). However, L may cancel out in the calculation, leading
  • #1
wee00x
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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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  • #2
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.
 
  • #3
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?
 
  • #4
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?
 
  • #5
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...
 
  • #6
Please write down your entire calculation then I can help you with it.
 

1. What is a particle-in-a-box model and how is it used in science?

A particle-in-a-box model is a simplified representation of a quantum system, where a particle is confined to a one-dimensional box. It is used in science to study the behavior and properties of quantum particles, such as electrons, in a confined space.

2. How does a particle-in-a-box model relate to the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum particles. The particle-in-a-box model is based on this equation and provides a simplified solution for a particle confined to a box.

3. What are the limitations of using a particle-in-a-box model?

A particle-in-a-box model is a simplified representation of a quantum system and does not accurately describe the behavior of particles in more complex environments. It also does not take into account the effects of interactions between particles.

4. How is the energy of a particle determined in a particle-in-a-box model?

In a particle-in-a-box model, the energy of a particle is determined by the size of the box and the wavelength of the particle. The allowed energy levels are quantized, meaning they can only take on certain discrete values.

5. What real-world applications can a particle-in-a-box model be used for?

A particle-in-a-box model has applications in various fields, such as materials science, where it can be used to understand the behavior of electrons in nanoscale materials. It is also used in quantum computing and in understanding the electronic properties of molecules in chemistry.

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