Matrix transmission coefficient

[amjad-sh]

TL;DR Summary

I'm working in a paper, in which the transmission coefficient $\hat t_k$ is written in a matrix form.
To check out if my work is right, I want to calculate $t_kt_k^*$. It must be less than 1.
The problem is that the transmission coefficient in the case I'm working on is in matrix form,it is not a number.

The Hamiltonian of the system I'm working on is in the form :
$\hat H=\dfrac{p^2}{2m}-\dfrac{\partial_z^2}{2m}+V(z)+\gamma V'(z)(\hat z \times \vec{\mathbf p})\cdot \vec{\sigma}$
There is translational symmetry in the x-y plane.
$\vec{\mathbf{p}}$ is the two dimensional momentum in the x,y plane.
$-\dfrac{\partial_z^2}{2m}$ is the kinetic energy in the z direction.
$\gamma V'(z)(\hat z \times \vec{\mathbf p})\cdot \vec{\sigma}$ is the spin orbit coupling correction term.
$V(z)=V\theta(z)$

The solution of this problem may be solved to be:

$\vec{\varphi_{k\sigma}}(z)= \begin{cases} (e^{ikz}+\hat r_ke^{-ikz})\chi_{\sigma} & \text{if } z < 0 \\ \hat t_ke^{ik'z}\chi_{\sigma} & \text{if } z> 0 \end{cases}$

according to the paper$\hat r_k$ and $\hat t_k$ are the matrix reflection and transmission coefficients respectively.

Where $\hat r_k=r_0\sigma_0 +\mathbf{\hat r} \cdot \vec{\sigma}$
and $\hat t_k=t_0\sigma_0 +\mathbf{\hat t} \cdot \vec{\sigma}$

and $\vec{\sigma}$ is the vector of pauli matrices.

$\mathbf{\hat r}$and $\mathbf{\hat t}$ are spin flip operators.

as you can see the transmission and reflection coefficients are in matrix form.
so what I should do in this case? can we generalize the transmission coefficient to be in matrix form?and how I can use it physically?
you can check the paper in the attachment file(second section).

 

[A. Neumaier]

To check out if my work is right, I want to calculate $t_kt_k^*$. It must be less than 1. [...]
as you can see the transmission and reflection coefficients are in matrix form.
so what I should do in this case? can we generalize the transmission coefficient to be in matrix form?

You can still form $t_kt_k^*$, using the conjugate transpose for the adjoint. It is Hermitian positive definite, hence its eigenvalues are real. The condition is now that all eigenvalues must be less than 1.