Linear Harmonic Oscillator Quantum Mechanics

[nrqed]

Homework Statement

A linear harmonic oscillator with frequency ω = h_bar / M is at time t = 0 in the state described by the wave-function:
Ψ(x,0) = C( 1 + √2x) e^-x2/2

Determine the values of energy which can be measured in this state.

I'm not really sure where to start this question and was wondering if someone could help me get the ball rolling.

Homework Equations

The Attempt at a Solution

The idea is to write the wave function given to you as a linear combination of the harmonic oscillator wave functions. By that I mean you write

$$\Psi(x,0) = C_0 \Psi_0(x,0) + C_1 \Psi_1(x,0) + C_2 \Psi_2(x,0) + \ldots$$

where the coefficients $C_0,C_1,C_2 \ldots$ are unknowns for now and by $\Psi_0(x,0), \Psi_1(x,0) \ldots$ I mean the wave functions of the harmonic oscillator corresponding to n=0, n=1, n=2, etc.

Now, your goal is to determine the values of the unknown coefficients $C_0,C_1,C_2 \ldots$. In fact, you need to do something simpler: just determine which ones are nonzero. Do you see how to do that? (hint: you must work with each power of x times the exponential separately.)