This one might take some brains !!! Harmonic Oscillator?

Question

I need find he coefficients C2 and C1 from the wavefunction
PSI(x,0) = C2 |2> + C1 |1>

for harmonic oscillator of mass m and frequency w

|2> and |1> are normalized eigen functions for n = 2 and n = 1

Find the coefficients C2 and C1 such that the expectation value for the total energy, for |PSI(x,0)> equals = 2(h bar)w

and also...
Write the expression for the time dependence of PSI(x,t)
Calculate the probability to find system in state |2> as a function of time

Calculate as a function of time
Calculate as a function of time

Whatever help is possible is GREATLY appriciated.

Answer 1

First of all, because these energy eigenstates have different eigenvalues, they are orthonormal functions, and the Pythagorean Theorem applies:
[1] ... psi^2 = C1^2 + C2^2 = 1
(we want psi to be normalized, right?)

Since, the eigenfunctions are
[2a] ... E1 = 3/2 hbar w
[2b] ... E2 = 5/2 hbar w

We also have
[3] ... = C1^2 E1 + C2^2 E2 = 2 hbar w

This gives the system of equations
[4a] ... 1/2 C1^2 + 3/2 C2^2 = 2 hbar w
[4b] ... C1^2 + C2^2 = 1
with obvious solutions C1^2 = C2^2 = 0.5. Therefore, C1 and C2 are equal to 1/sqrt 2 (up to a phase factor exp(i phi); I will assume the phase angles are zero).

Time dependent:
[5a] psi(x,t) = C1 exp(-i E1 t / hbar) |1> + C2 exp(-i E2 t / hbar) |2>
[5b] ... = (1/sqrt 2) {exp(-3/2 iwt) |1> + exp(-5/2 iwt) |2>}