- #1

amjad-sh

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- 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).

##\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).