Reflection and Transmission Matrix for a 2x2 Hamiltonian in a Conducting Channel

[sokrates]

Suppose I have a 2x2 Hamiltonian describing a conducting channel, as a black box.

$$H_{Rashba}=\sigma_y k_x - \sigma_x k_y$$

and also suppose I have an incident 2x1 wavefunction with up and down spin components, to this black box.

What I am trying to do is to write a 2x2 reflection (or transmission) matrix for the wavefunction that is incident to this region.

How can I do this? How do you go about doing such a problem? What are the methods or appropriate terminology for this kind of problem?

I hope the question is clear... Thanks in advance for any guidance.

(This is not homework) ( and I don't want the solution, just the method)

 

[sokrates]

There are so many epistemologists in this forum, nowadays, that nobody can answer even the simplest physics problems.

Good luck figuring which interpretation is better,

I found my answer.