Electron in a box, superposition of first three energy eigenstates I have a normalised waveform:
$$\psi (x,t) = \frac{1}{\sqrt{3}}u_{1}(x) + \frac{1}{\sqrt{3}}u_{2}(x) + \frac{1}{\sqrt{3}}u_{3}(x)
$$
where the $u$'s represent energy eigenstates. The electron is in a one-dimensional box where $0<x<a$.
I have been asked to find the frequencies associated and write down explicitly the wavefunction with respect to position $x$ and time $t$.
I got the following when I tried this myself:
$$\psi (x) = \frac{1}{\sqrt{3}}u_{1}(x)e^\frac{-i\hbar\pi^2t}{2ma^2} + \frac{1}{\sqrt{3}}u_{2}(x)e^\frac{-i4\hbar\pi^2t}{2ma^2} + \frac{1}{\sqrt{3}}u_{3}(x)e^\frac{-i9\hbar\pi^2t}{2ma^2}$$
I am really looking to just know if this is actually what I am being asked for?
Also to prove/calculate its periodicity is it just simply a case of 
$$\psi (x) = \psi (x+t)$$
Sorry for any confusion. I am just looking to make sure this is being done right.
I hope the question is okay; it's my first time posting
 A: Look at the  $\psi (x,t)$ you've found:

$$\psi (x) = \frac{1}{\sqrt{3}}u_{1}(x)e^\frac{-i\hbar\pi^2t}{2ma^2} +
 \frac{1}{\sqrt{3}}u_{2}(x)e^\frac{-i4\hbar\pi^2t}{2ma^2} +
 \frac{1}{\sqrt{3}}u_{3}(x)e^\frac{-i9\hbar\pi^2t}{2ma^2}$$

This is absolutely correct. Now to find the periodicity all you need to find is that special time T for which $\psi (x)$ goes back to it's initial configuration. In your case, i.e. one dimensional box, all energies are some multiple of ground state energy. $$E_n = n^2 E_0$$ $E_0$ being ground state energy. And you've already found the frequencies to be:
$$f = E_n /\hbar$$ for n-th state. Thus all frequencies will be multiples of the frequency associated with ground state.
And since time period is $\frac{1}{f}$ that implies that ground state has largest time period. In the time it comes back to it's original setting all other states have also come back to their exact initial condition. Thus your wavefunction's time period is the time period of ground state eigenvector itself. Namely,
$$T = \frac{2ma^2}{\pi^2 \hbar}$$.
