Particle in a Box, normalizing wave function

[EbolaPox]

Question from textbook (Modern Physics, Thornton and Rex, question 54 Chapter 5):

"Write down the normalized wave functions for the first three energy levels of a particle of mass m in a one dimensional box of width L. Assume there are equal probabilities of being in each state."

I know how to normalize a wave function, I'm just not too sure exactly how I do so for three different energy levels.

My work and ideas:

Previously in the chapter, they stated $$\Psi = A sin(kx - \omega t)$$. Next, I saw $$E_n = \frac{ \hbar ^2 n^2}{8ml^2}$$. I know for the wave function to be normalized, I need $$1 = \int_{0}^{l} |\psi|^2 dx$$, but I'm not too sure how to proceed or really use any of this. Do they want me to somehow algebraically manipulate my kx-wt to get an expression for energy inside? Perhaps I'm just confused as of to what they want me to do. Any hints or pointers in the right direction would be great. Thank you.

What really annoys me is I've done the problems in the next chapter and some problems in Griffiths intro to QM book, but this problem which seems so simple is the only one that has a solution that has evaded me. Thanks again for any help.

 

[Ahmes]

Well, you've got it wrong.
What is $$\Psi$$? Usually* this is the solution to $$\mathcal{H} \Psi = E_n \Psi$$. So it shouldn't be time dependent. For a particle in a box you should solve just note the boundary conditions - zero at the ends of the box, so $$\Psi_n \propto \sin \left( \frac{n\pi x }{L} \right)$$
To normalize, you should do what you said that should be done - write $$1 = A^2 \int_{0}^{l} |\psi|^2 dx$$ where $$\psi$$is what we found before, and find A. So $$\Psi=A\psi$$.
The answer is that the normalization factor is always √2/√L

Sometimes $$\Psi$$ is said to be the solution to the time dependent Schrödinger Equation, the difference for your problem is just a global phase.

 

[EbolaPox]

Ah! Thank you very much. That makes sense. I was just wondering if I could write $$\Psi$$ in that form, but was worried that I had to have a time dependence in there.