Probability of a particle in a box in the first excited state.

[kickingpaper]

Homework Statement

A particle is in the first excited state of a box of length L. Find the probability of finding the particle in the interval ∆x = 0.007L at x = 0.55L.

Homework Equations

P = ∫ ψ*ψdx from .543L to .557L

The Attempt at a Solution

Normalizing ψ gives ψ=√(2/L)sin(nπx/L)
P = ∫ ψ*ψdx = ∫(2/L)sin^2(nπx/L)dx from .543L to .557L
The integration simplifies to
P = x/L - sin(4πx/L)/4
so P = [.557L/L - sin(4π*.557L/L)/4] - [.543L/L - sin(4π*.543L/L)/4]
P = 0.0132 or 1.32%

This is wrong though and the hint given afterwords was that because the Δx is so small, there is no need for integration. This just confuses me because abs(ψ)^2 will have a 1/L factor in it. Any help will be useful. Thanks!

 

[BruceW]

Ah, ok, think of how you would approximate the area under a small section of a graph. (What is the simplest shape you can use?) And your 'graph' is abs(ψ)^2 against x, with Δx= 0.007L

 

[kickingpaper]

Thanks

Thanks for reminding me that dx can be approximated as Δx. For some reason I didn't make that jump

 

[BruceW]

hehe yeah, that's alright. It is quite unusual to do an 'approximate integral' in this way. You could maybe even calculate the fractional error (to first order), by calculating the derivative of abs(ψ)^2, and finding the difference that this makes to the approximation.

(but that's not part of the question, so whatever).

 

[Nickie]

I would like to point out that the given information is not enough to accurately determine the probability of finding the particle in the specified interval. The probability depends on the specific values of n and L, which are not provided in the given content. Additionally, the equation for normalizing ψ is incorrect. It should be ψ = √(2/L)sin(nπx/L) instead of ψ = √(2/L)sin(nπx).

Assuming that the equation for normalizing ψ is corrected and the values of n and L are known, then the probability can be calculated by simply squaring the value of ψ at x = 0.55L. This is because the probability density function is given by ψ*ψ and the interval ∆x = 0.007L is small enough that it can be approximated as a point. Therefore, the probability of finding the particle at x = 0.55L is given by P = |ψ(0.55L)|^2.

In general, it is important to provide all necessary information and use correct equations when solving problems in physics. Without this, the results may be inaccurate and misleading.