Quantum Mechanics Superposition Question

[jumbogala]

Homework Statement

A particle in an infinite square well has as its initial wavefunction ψ(x,t=0) = (1/√2)[φ₁ + φ₃].

Find |ψ(x,t)|². Express it as a sinusoidal function of time using ω= π²*h/(2mL²).

Homework Equations

Note that φ_n = √(2/L)*sin(nπx/L) for a well of width L.

The Attempt at a Solution

I found something in my book which says that ψ(x,t) = Ʃb_n*e^-iω_ntφ_n.

Therefore, my ψ(x,t) = (1/√2)[e^-iω₁t)φ₁ + e^-iω₃t)φ₃].

Is that right so far?

Then, I can compute |ψ(x,t)|², but I don't see how that is going to give me an answer in terms of ω. I have ω_n though.

 

[vela]

How are $\omega$ and $\omega_n$ related to the energy of state n?

 

[genericusrnme]

When you say you found something in your book that says $\Psi (x,t) = \sum_n b_n\ e^{-i\ \omega _n \ t}\ \psi _n(x)$, do you know what that is?
Also, do you know what $\omega _n$ actually is?