31.32 please answer

The wave function at time t = 0 for a particle in a harmonic oscillator potential V= - kx2, is of the form 4 (x, 0) = Ae -(az)'/2 cos B Ho (aI) + sin B 2 V2 H2 (az) where S and A are real constants, o'= vmk/h, and the Hermite polyno- mials are normalized so that [Hx(ar )]2 dx = 2" n! . (a) Write an expression for v (x, t). (b) What are the possible results of a measurement of the energy of the particle in this state, and what are the relative probabilities of getting these values? (c) What is (x) at t = O? How does (x) change with time?(a) For a particle of mass m in a one-dimensional harmonic oscillator potential V = mwar?/2, write down the most general solution to the time- dependent Schrodinger equation, (r,t ), in terms of harmonic oscillator eigenstates on (I). (b) Using (a) show that the expectation value of I, (x), as a function of time can be written as A cos wt + B sin wt, where A and B are constants. (c) Using (a) show explicitly that the time average of the potential energy satisfies (V) = (E) for a general (I,t). Note the equality mw n+1 ron = h 2 On+1 + On-1