(Solved): Consider the particle in a 1D-box, whose wave function is given by: (x)=(L2 ...

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Consider the particle in a 1D-box, whose wave function is given by: $$?(x)={\left(\frac{2}{L}\right)}^{\frac{1}{2}}\sin \left(\frac{n?}{L}x\right)\mathrm{n}=1,2,3\dots 0<\mathrm{x}<\mathrm{L}$$ a) [3] Show that the wave function is a solution to the Schrodinger equation for all ' $$n$$ '. b) [3] Show that the wave function is normalized for all ' $$n$$ '. c) [5] Derive an expression for the probability of finding the particle from $$x=L/4$$ to $$L/2$$ (from the first quarter of the box to the middle of the box) for any state $$\mathrm{n}$$. What is the numeric value of probability for the ground state? d) [4] Determine the expectation value $$?{\hat{p}}_x^2?$$. e) [3] What is the variance in the measurement of $$p_x$$ ? NOTE: for parts d and e, you can argue on physical grounds that certain expectation values may have a particular value as long as you provide a clear argument.