Slides: The finite well

This lecture is interspersed into the activity. The main point is to solve the energy eigenvalue equation in a potential energy well that is piecewise continuous, matching the boundary conditions.

Discuss the consequences of for physical interpretation if the wave function or its derivative were not continuous (double valued probability density, undefined kinetic energy).

Use whatever form (exponential, sinusoidal, combination) of the wave function pieces seems intuitive to the students. It is usually best to keep the unknown energy $E$ explicit and the to keep the potential energy in each region explicit until the students request a simplifying notation $${k_{1(2,3)}} = \sqrt {\frac{{2m}\left(E-V_{1(2,3)}\right)}{\hbar ^2}} $$. If you introduce $k$ too soon, students may not recognize the $k$ values as merely a reparameterization of the unknown $E$, and may think that three new variables have been introduced!

Discuss the boundary conditions at infinity that lead to zero values of the coefficients of the terms that blow up at infinity. Discuss the “internal” boundary conditions. Discuss the fact that normalization gives another constraint. Explicitly count off the the number of unknowns against the number of constraints.

Discuss graphical solutions of the constraint equations.