1. At t=0, a linear harmonic oscillator is in a state described by the normalized wave function (x,0)=51u0(x)+21u2(x)+c3u3(x) where un(x) is the n-th eigenfunction of the corresponding Hamiltonian. (a) Determine the value of c3 assuming that it is real and positive. (b) Write the wave function for t>0. (c) What is the expected value of the oscillator energy at t=0 ? And at t=1s ? (d) What is the dispersion of this mean value? (e) At t=t0, the energy of the system is measured. Which are the possible outcomes of this measurement and what is the probability of obtaining them? (f) Suppose that at t=t0, the result of the measurement is /2. What is the state of the system after this measurement? (g) The energy is measure later at t=2t0. Which are the possible outcomes of this measurement and the corresponding probabilities