Particle in a potential well of harmonic oscillator

[Roodles01]

Homework Statement

I have a similar problem to this one on Physicsforum from a few years ago.

Homework Equations

Cleggy has finished part a) saying he gets the answer as
Ψ(x, t) = (1/√2) (ψ¹(x)exp(-3iwt/2+ iψ₃(x)exp(-7iwt/2)

OK
classical angular frequency ω₀ = √C/m for period of oscillation T = 2∏ / ω₀
I note that E₀ = ½ ħ ω₀

The Attempt at a Solution

I have, for a wave packet with equal coefficients for 1st & 2nd stationary state wave function;
ψA(x,t) = 1/√2 (ψ0(x) e^-iwt/2 + ψ1(x) e^-3iwt/2)

This question asks for vaidity for t>0

I don't get how he got there.
Could someone expand this please

 

[Thaakisfox]

Suppose $$\psi_n(x,t)$$ are the eigenstates with energy $$E_n$$.

Then a generic state (general solution of the Schrödinger equation) can be written as the superposition of these states:

$$\psi(x,t)=\sum_{n}a_n\psi_n(x,t)=\sum_{n}a_n\psi_n(x)e^{-iE_n t/\hbar}$$

Now match this with the initial condition to find the coefficients a_n and use the energy states for the harmonic oscillator:

$$E_n=\hbar\omega\left(n+\frac12\right)$$

 

[Roodles01]

Ah! Yes, I hadn't done reading ahead enough, so hadn't got to that bit.
Thanks for the pointer, though.