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

AI Thread Summary
To find the probability of a particle in the first excited state within a box of length L, the probability is calculated using the wave function ψ normalized to the box dimensions. The integration of the probability density function over the small interval ∆x = 0.007L from 0.543L to 0.557L yields a result of approximately 1.32%. However, due to the small size of ∆x, it is suggested that integration may not be necessary, and the area can be approximated directly. The discussion emphasizes the importance of recognizing that dx can be treated as Δx for small intervals, simplifying the calculation. Understanding this approximation can clarify the confusion surrounding the integration process.
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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!
 
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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
 
Thanks

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