Say I have a potential that is infinite from - infinity to 0, -V_0 from 0 to a, and 0 from a to infinity.
Then I have three unknowns, two for the plane wave solution in the finite well and one for the exponential outside the well.
But I also have four equations. Two wavefunction continuity boundary conditions and a wavefunction derivative continuity boundary conditions. I also have the normalization equation!
How is this possible? How can the solutions depend on which 3 of the four equations I select to use?