3. (5 points) A particle of mass m is in one-dimensional harmonic oscillator potential V(a) = =mwar. The initial wave function for this particle at time t = 0 is given by a linear combination of two stationary states V(x, t=0) = A -V740 + 241 .Here wo, 1 are the (real) eigenfunctions of the time-independent Schrodinger equa- tion for harmonic oscillator corresponding to the ground state and first excited level respectively. (a) Normalize this wave function. (b) System evolves in time according to the time-dependent Schrodinger equation. Construct the wave function at time t > 0, writing down the explicit form of the energies. (c) Construct the probability density at time t > 0. (d) What is the expectation value of the energy at time t ? Express in terms of w. (e) What are the probabilities that the measurement of energy will give the result for the ground, first and second excited level respectively