Consider the ansatz b1(x) = ax o(x) for the first excited state of the harmonic oscillator. determine the positive constant a and the energy eigenvalue e1. check dimensions. check that the result is compatible with the general formula en for the spectrum of the harmonic oscillator. (n+)has b.4 10 p. we prepare the state (x) o(x)+1(x)) with o(x) and (x) the ground state and first excited state found above. what is the probability p(r)d that, in a measurement of the position, we find the particle in the interval r, xdr]? plot p(x) b.5 [10 p. we prepare the state (x) the particle. what is the outcome of the position averaged over many preparations and (ar)) and measure the position à of measurements? 1 b.6 [10 pl. at the time to o(r)(r)) and let it 0, we prepare the state bto (x) evolve isolated up to the time t1, when we make a measurement of the energy h. what is the probability of finding energy hw? explain (o( and let it b.7 10 pl. at the time to 0, we prepare the state t ( evolve isolated up to the time t1, when we make a measurement of the momentum p. = if we repeat preparation, evolution and measurement many times, what is the average outcome of the momentum at the time t1? compute the result and compare it to what the ehrenfest theorem tells you about the evolution of the average momentum