There are mathematical solutions to the Schrödinger equation for the finite well for any e...

There are mathematical solutions to the Schrödinger equation for the finite well for any energy, and in fact, they can be made smooth everywhere. Guided by A Closer Look: Solving the Finite Well, show this as follows:

(a). Don't throw out any mathematical solutions. That is, in region II (x<0), assume that ψ(x) = Ce+ax + De-ax and in region III (x > L), assume that ψ(x) = Fe+LX + Ge-ax. Write the smoothness conditions.

(b). In Section 5.6, the smoothness conditions were combined to eliminate A, B, and G in favor of C. In the remaining equation, C canceled, leaving an equation involving only k and α, solvable for only certain values of E. Why can't this be done here?

(c). Our solution is smooth. What is still wrong with it physically?

(d). Show that

and that setting these offending coefficients to 0 reproduces quantization condition (5-22).