How Can Transfer Matrix Method Automate Phase Matching in Quantum Mechanics?

[IHateMayonnaise]

Just for fun I'm doing a graphical representation of a QM problem out of Merzbacher. Basically we have a regular old potential barrier, with incoming particle of energy E>Vo. I am showing that as you increase the energy, the transmission is increased, i.e. the wavefunctions in all three areas match.

Anyway..I'm not sure how to go about matching the wave functions at the boundary, I know all I have to do is change the phase of each wave fn but that's a pain, and I want this to be as automated as possible. Please see pdf of my plot, code can be shown if requested.

Any ideas?

 

[NateD]

One possible solution is to use the transfer matrix method. The transfer matrix method is a technique used to calculate the transmission of a wave through a system of multiple layers and interfaces. By using the matrix method, you can calculate the transmission coefficient for each layer, as well as the overall transmission coefficient for the entire system. This can be used to match the wave functions at the boundary. You can set up a transfer matrix for each layer in the system. The elements of the transfer matrix depend on the properties of the layer, such as material type, thickness, etc. The transfer matrix is then multiplied together to obtain the overall transmission coefficient. The overall transmission coefficient can then be used to adjust the phase of the wave functions at the boundary, allowing them to match.