Square Well boundary conditions.

This video shows how to use boundary conditions along with the time-independent Schrodinger equation
to obtain the wave-function parameters for the bound states of a finite square well. Normalizing and actually matching the wave-functions at the boundaries, takes more work.

At 2:00 I say " the states will divide up into even and odd states",
but really I should have said "the energy eigenstates will divide up into even and odd states".

Added notes, 4-17: So basically the procedure you can use is:
1) obtain k as outlined in the video
2) use k to obtain b
3) use k to obtain B/A
4) use the normalization condition to obtain specific numerical values of A and B. That condition is that the integral from -infinity to infinity must be 1, so that the sum of three integrals for the three different regions. That sum must add up to one.

Added note, 4-17, 3:45 PM 
The way one gets k in the video is totally fine, but there are lots of ways. All starting from the same place. For example, you can use: $k^2+b^2=2mU/\hbar^2$, from the Schrodinger equation, and $ktan(k)=b$ from the boundary conditions at x = 1nm.  Feed those to Wolfram alpha and it will give you values of k and b which "belong to" particular energy eigenstates. This way is kind of appealing since you have a nice circle as part of your graph, with a radius proportional to U, and the values of b and k come together in pairs.
(In general it is: $ktan(kL/2)=b$. For our particular case, L/2=1 nm.)