A particle is in one-dimensional harmonic oscillator potential V (x) = 1/2 mω²x², where m is mass of the particle and ω is the frequency. The initial wave function for this particle at time t = 0 is given by a linear combination of two stationary states:
Ψ(x, t = 0) = A [3e^(iΦ₁)Ψ₁ + √7 Ψ₂]
Here Φ₁ is given real constant and Ψ₁, Ψ₂ are the (real) eigenfunctions of the time-independent Schrödinger equation for harmonic oscillator corresponding to the first and second excited level respectively.
(a) Compute the expectation value of position in state Ψ(x, t).
(b) Compute the expectation value of momentum in state Ψ(x, t) (explicit calculation, not Ehrenfest theorem).
(c) Verify that Ehrenfest theorem is satisfied.
(d) At time T, the measurement of energy was performed, and the result was E2. What is the particle wave function at time t >T?

For this problem use ladder operators and orthonormality condition for eigenfunctions. There is no need to write down explicit formulae for wave functions with Hermite polynomials.