Find Result <x> for Particle in n=3 Excited State of Rigid Box

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In summary, the expectation value for the position of a particle in the second excited state of a rigid box is equal to (2/a) multiplied by the integral of x multiplied by sin^2(n*pi*x/a) evaluated from 0 to a, where n represents the state of the particle. The answer for the ground state (n = 1) is a/2 if the boundaries of the box occur at 0 and a. To solve the integral, it may be helpful to use an integral-solver software application.
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SpaceTrekkie
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Homework Statement


find the result <x> (the expectation value) found when the position of a particle in the second (n=3) excited state of a rigid box.


Homework Equations


<x> = (2/a)[tex]\int[/tex]xsin2(xpi/a)dx

evaluated from 0 to a



The Attempt at a Solution



well when that integral is evaulated using the ground state the answer is a/2. I am not sure how they even got that. But my real problem is, is where does the n = 3 come in?

and where do I go from there?

thanks...
 
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  • #2
"n" appears in the numerator of the argument of the sine-squared term. "n" denotes the state of the particle.

a/2 for the ground state (n = 1) is correct only if the boundaries of the box occur at 0 and a. To get the answer, all you need to do is integrate. In QM, the integrals may get a little complicated to do by hand, so a good integral-solver software application (e.g. Maple) will likely benefit you.
 
  • #3
ooo ok. Thanks, I figured it out. The use of the boundary conditions are what was confusing me.
 

1. What is the "n=3 excited state" of a particle in a rigid box?

The n=3 excited state refers to the third energy level of a particle in a rigid box. In quantum mechanics, it is the level of energy that the particle possesses when it is in a confined space with specific boundary conditions.

2. What is a rigid box in the context of this experiment?

A rigid box is a theoretical model used to represent a confined space in which a particle can move freely. It is often used in quantum mechanics to study the behavior of particles in confined spaces.

3. How is the result for a particle in n=3 excited state of a rigid box determined?

The result for a particle in n=3 excited state of a rigid box is determined through mathematical calculations using the Schrödinger equation. This equation takes into account the properties of the particle, the dimensions of the rigid box, and the boundary conditions to calculate the energy levels of the particle.

4. What is the significance of finding the result for a particle in n=3 excited state of a rigid box?

Finding the result for a particle in n=3 excited state of a rigid box is significant because it allows us to better understand the behavior of particles in confined spaces. This knowledge can be applied in various fields, such as quantum computing and nanotechnology.

5. Can the result for a particle in n=3 excited state of a rigid box be experimentally verified?

Yes, the result can be experimentally verified through techniques such as spectroscopy, which can measure the energy levels of particles in confined spaces. These experiments provide evidence for the accuracy of theoretical calculations and help validate our understanding of quantum mechanics.

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