(Solved): Problem 3: Simple Harmonic Oscillator (total of 30 points) = Consider a particle of mass m in a si ...

Problem 3: Simple Harmonic Oscillator (total of 30 points) = Consider a particle of mass m in a simple harmonic oscillator (SHO) potential V(x) mw²x². At t=0 the particle is in the state 1 v(x,0) = Co (uo(x) — —?2u?(x), (2) where the un(x) are the usual normalized stationary state wavefunctions for the SHO, with up the ground state, and Co is a constant. a) (10 points) Calculate the expectation value (x) as a function of time (please be sure to normalize (x, 0)). b) (20 points) At t = T the particle is in the state V(x, 7) = C? (U?(a) — —2ª1(x)), (3) 2 where C? is a constant. What is the smallest value of T?