Let w,,(x) denote the orthonormal stationary states of a system corresponding to the energy E- Suppose that the normalized wave function of the system at time = 0 is w(x, 0), and suppose that a measurement of the energy yields the value E₁ with probability 1/2, E2 with probability 3/8, and E3 with probability 1/8. (a) Write the most general expansion for w(x, 0) consistent with this information? (b) What is the expansion for the wave function of the system at time 1, y(x, t)? (c) Show that the expectation value of the Hamiltonian does not change with time. (d) Calculate the probability density, p(x, f), and the probability current density, J(x, t). Verify that the probability is conserved; that is, show that op/t + (x,t) = 0. Consider a neutron which is confined to an infinite potential well of width a = 8 fm. At time 1 = 0 the neutron is assumed to be in the state 'T(x, 0) sin ( □()+厚 2xx' sin + sin (2) (a) If a measurement is carried out on the system, what are the values that will be found for the energy and with what probabilities? Express your answer in MeV (the mass of the neutron is mc2 939 MeV, he 197 MeV fm). (b) If this measurement is repeated on many identical systems, what is the average value of the energy that will be found? Again, express your answer in MeV. (c) Using the uncertainty principle, estimate the order of magnitude of the neutron's speed in this well as a function of the speed of light e. Consider the dimensionless harmonic oscillator I lamiltonian d with i dx (a) Show that the two wave functions yo(x) = x²/2 and wi(x) = xe²/2 are eigenfunc- tions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the value of the coefficient a such that w2(x) = (1+ax²) ex²/2 is orthogonal to yo(x). Then show that w2(x) is an eigenfunction of Ĥ with eigenvalue 5/2.