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Question

A wave function of a particle with mass m is given by

$\psi(x) = \left\{ \begin{array}{ll} A ~ \cos{(\alpha x)} , & - \dfrac{\pi}{2\alpha} \leq x \leq \dfrac{\pi}{2\alpha} \\ 0 , & \text{Otherwise.} \end{array}\right.$

where $\alpha = 1.00 ~ \times 10^{10} ~ \mathrm{m}^{-1}$. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0 \leq x \leq 0.5 ~ \times 10^{-10} ~ \mathrm{m}$. (c) Find the particle’s average position. (d) Find its average momentum. (e) Find its average kinetic energy $-0.5 ~ \times 10^{-10} \mathrm{m} \leq x \leq 0.5 ~ \times 10^{-10} ~ \mathrm{m}$.

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