Basic QM - Trying to catch up in a class

[FrogPad]

I haven't taken one of the recommended courses for a class I am in, so I'm playing a little catch-up here.

Q: The wave function of an electron in a one-dimensional infinite square well of width a,x -> (0,a) at time t=0 is given by:

$$\psi (x,0) \sqrt{\frac{2}{7}}\psi_1(x) + \sqrt{\frac{5}{7}}\psi_2(x)$$

where $$\psi_1(x)$$ and $$\psi_2(x)$$ are the ground and first excited stationary state of the system respectively, $$\psi_n(x)=\sqrt{\frac{2}{a}}\sin (n\pix/a), \,\,\, E_n = n^2\pi^2 \bar h^2/(2ma^2), \,], n=1,2,...$$

a) Write down the wavefunction at time t in terms of $$\psi_1(x)$$ and $$\psi_2(x)$$.

b) You measure the energy of an electron at time t=0. Write down possible values of the energy and the probability of measuring each.

c) Calculate the expectation value of hte energy in the state $$\psi(x,t)$$ above.

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I am lost here. I am stuck on (a) right now, and really do not know where to go. To me it looks like $$\psi_n(x)$$ is the wavefunction already solved from the Schrödinger equation (with the boundary conditions 0 and a). So I need to express the total wavefunction in terms of two states psi_1, and psi_2. This to me seems like another condition that would come from a differential equation something with a time dependence. I am lost...

Would someone nudge me in a proper direction?

thanks,

 

[dextercioby]

Hint for 1. $$\psi (t)=U(t)\psi=e^{\frac{1}{i\hbar}tH} \psi$$

Daniel.