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(a) State Born's rule for a particle in one dimension, and explain why this requires the wave function to be normalized. (b) A particle is described by the wave function V (7,0) = 6 e /2a? πα where a is a positive constant. Calculate the probability of finding the particle somewhere in the region from x 0 to x = +a. You may use the standard integral for e du = 0.747. 3 (a) At time t = 0, a one-dimensional bound system is in a state described by the normalized wave function V (1,0). The system has a set of orthonormal.energy eigenfunctions y, (), 42(x).... with corresponding eigenvalues E1, E2..... Write down the overlap rule for the probability of getting the energy E, when the energy is measured at time (= (b) Suppose that a system is described by a normalized wave function of the form \(x,0) = anun(x). 2 where the an are complex constants. Prove that ai Livm. 41(2) V (2,0) dr, including all the steps in your working.