Word Count : 361

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To calculate the expectation values \(\langle \hat{x} \rangle\) and \(\langle \hat{x}^2 \rangle\) for the given wave function \(\psi_n(x)\), we need to evaluate the following integrals. 1. Expectation value of \(\hat{x}\): \[ \langle \hat{x} \rangle = \int_{0}^{a} \psi_n^*(x) \, x \, \psi_n(x) \, dx \] Since the wave function \(\psi_n(x)\) is real and symmetric, we have: \[ \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right) \] Thus, \[ \langle \hat{x} \rangle = \int_{0}^{a} \left( \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right) \right) x \left( \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right) \right) dx \] Simplifying: \[ \langle \hat{x} \rangle ______ ______ _______ ______ _________ ______.
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