**Text Explanation for Educational Website**---**Quantum Mechanics: Particle in a One-Dimensional Box**Consider a particle in the $ n = 1 $ state within a one-dimensional box of length $ a $, with infinite potential at the walls. The normalized wave function for this system is given by:\[\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right)\]**Task:**(a) Calculate the probability of finding the particle between $ \frac{a}{2} $ and $ a $.*Hint: It may be helpful to visualize the box and sketch the probability density.***Graphical Explanation:**The wave function $\psi(x)$ describes the quantum state of the particle in terms of its position $x$ within the box. In this scenario:- The graph of $\psi(x)$ is a sine wave that starts at 0, increases to a maximum, and then decreases back to 0 from $x = 0$ to $x = a$.- The probability density, given by $|\psi(x)|^2$, represents the likelihood of finding the particle at a certain position $x$. In this case, it will be a squared sine wave, indicating regions where the particle is more or less likely to be found.To calculate the probability, integrate $|\psi(x)|^2$ from $\frac{a}{2}$ to $a$.---

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Answered: Consider a particle in the n = 1 state…

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