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Hennessy

- 22

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**TL;DR Summary:**Looking for help on a Intro to QM Problem

Hi All, THIS IS A GRADED PIECE OF WORK AT MY UNIVERSITY PLEASE DO NOT JUST GIVE ME THE ANSWER , I have made this post to see if what i've calculated seems reasonable, it sounds unlikely as 0.4 - 0.5L is in the middle of the well. The context is that we are in an infinite potential square well. The Question is : A quantum particle is trapped in a 1D infinite well with length L. Find the probability of finding the particle between 0.4L and 0.5L for the ground state. My answer is as follows:

$$ \psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi n x}{L}\right)$$ satisfies a free particle in a infinite square well. For the ground state we have $$ \psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi n x}{L}\right)$$ where n = 1 for the ground state and so becomes $$ \psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right)$$ Now we need to calculate $\lvert(\psi_x)\rvert^2$ which is $\psi^2(x) ={\frac{2}{L}} \sin^2\left(\frac{\pi x}{L}\right)$ Then to find the probability between 0.4 and 0.5 we need to evaluate this integral between those limits.

$$ \int_{0.4L}^{0.5L} (\psi_x)^2 dx = \frac{2}{L}\int_{0.4L}^{0.5L} \sin^2\frac{\pi x}{L} dx$$ Since $\sin^2$ cannot be integrated directly without producing Gaussian error functions we use the trigonometric identity that $\sin^2(Ax) = \frac{1 - \cos(2Ax)}{2}$ Where $A = \frac{\pi }{L}$ This means that our integral becomes $$ \frac{2}{L}\int_{0.4L}^{0.5L} \frac{1 - \cos(2\pi x)}{2} dx$$ Pulling out the $\frac{1}{2}$ leaves us with $$\frac{1}{L}\int_{0.4L}^{0.5L} 1-\cos(2 \pi x) dx$$ Computing this integral $$\frac{1}{L} \left[ x - \frac{L}{2\pi} \sin\left(\frac{2\pi x}{L}\right) \right]_{0.4L}^{0.5L} $$

Computing for the upper limit leads us to $$ \frac{1}{L} \left[ 0.5L - \frac{L}{2\pi} \sin(\frac{2\pi L}{2L})\right] = 0.5 -\frac{2\pi}{1} \times \sin(\pi) = 0.5$$ For the lower limit $$= 0.4 - \frac{L}{2 \pi} \sin(0.8 \pi) = 0.4$$ so the probability that you will find the particle in the ground state between our 2 limits is $$ 0.5-0.4 = P = 0.1$$

Am i on the right tracks or have i made a mistake somewhere ? i think i'm correct but just want to have others opinion on if i have done something wrong and if i have please tell me where and why im wrong so i can improve myself in physics, thank you <3.

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