[UPSC CSE 2024] Foundations of Quantum Mechanics (Q7, 10M)

A particle limited to the x-axis has the wave function $\phi(x) = bx^2$ between $x = 0$ and $x = 2$ ; the wave function $\phi(x) = 0$ elsewhere. (i) Find the probability that the particle can be found between $x = 1.0$ and $x = 1.5$ . (ii) Find the expectation value $<x>$ of the particle position.