A harmonic oscillator is in a state such that a measurement of the energy would yield either (1/2) ̄hω or (3/2) ̄hω, with equal probability. (a) Write down the properly normalized time-dependent wave function which describes this state. (b) Solve for the expectation value of position as a function of time, ⟨x(t)⟩. What is the largest possible value of ⟨x(t)⟩ in such a state?

A harmonic oscillator is in a state such that a measurement of the energy would yield either (1/2) ̄hω or (3/2) ̄hω, with equal probability. (a) Write down the properly normalized time-dependent wave function which describes this state. (b) Solve for the expectation value of position as a function of time, ⟨x(t)⟩. What is the largest possible value of ⟨x(t)⟩ in such a state?