# Matrix transmission coefficient

• A
In summary, the Hamiltonian of the system described in the conversation includes a kinetic energy term in the z direction and a spin orbit coupling correction term. The solution to this problem involves matrix reflection and transmission coefficients, which can be used to calculate the transmission coefficient in matrix form. To ensure the accuracy of the solution, the eigenvalues of the resulting matrix must be less than 1.
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).

#### Attachments

• paper.pdf
2.6 MB · Views: 151
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.

## 1. What is a matrix transmission coefficient?

A matrix transmission coefficient is a mathematical value that represents the ratio of the transmitted wave amplitude to the incident wave amplitude in a system described by a matrix. It is commonly used in quantum mechanics to calculate the probability of a particle passing through a potential barrier.

## 2. How is the matrix transmission coefficient calculated?

The matrix transmission coefficient is calculated by taking the absolute value squared of the ratio of the transmitted wave amplitude to the incident wave amplitude. This can be represented by the equation T = |t|^2/|i|^2, where t is the transmitted wave amplitude and i is the incident wave amplitude.

## 3. What does a high matrix transmission coefficient indicate?

A high matrix transmission coefficient indicates that there is a high probability of a particle passing through a potential barrier. This means that the barrier is relatively transparent to the particle and it is likely to be transmitted without being significantly affected by the barrier.

## 4. How does the potential barrier affect the matrix transmission coefficient?

The potential barrier affects the matrix transmission coefficient by changing the values of the transmitted and incident wave amplitudes. This, in turn, affects the ratio and ultimately the probability of transmission. A higher potential barrier will result in a lower matrix transmission coefficient, indicating a lower probability of transmission.

## 5. Can the matrix transmission coefficient be greater than 1?

No, the matrix transmission coefficient cannot be greater than 1. This is because it represents a probability and probabilities cannot exceed 1. A value greater than 1 would imply that the probability of transmission is greater than 100%, which is not possible.

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