- #1

- 5

- 0

## Main Question or Discussion Point

Hi, I have a problem identifying some 3d rotation matrices. Actually I don't know if the result can be brought on the desired form, however it would make sense from a physics point of view. My two questions are given at the bottom.

[tex]\mathbf{s}=\left(

\begin{array}{c}

s_{x} \\ s_{y} \\ s_{z}

\end{array} \right),\; \mathbf{S}=\left(

\begin{array}{c}

S_{x} \\ S_{y} \\ S_{z}

\end{array} \right)[/tex]

[tex]H=\left[\left(

\begin{array}{ccc}

1-\left(1-C_{z}^{2}+C_{x}^{2}\right)\gamma^{2}& 0 & -2\gamma\left(1-C_{x}C_{z}\gamma\right) \\

0 & 1-\left(1+C_{z}^{2}+C_{x}^{2}\right)\gamma^{2} & 0 \\

2\gamma\left(1-C_{x}C_{z}\gamma\right)& 0 &

1-\left(1+C_{z}^{2}-C_{x}^{2}\right)\gamma^{2}

\end{array} \right) \mathbf{s}\right]\cdot\mathbf{S}

[/tex]

(for the physics interested: this describes Kondo effect in a quantum dot with spin-orbit interaction)

The goal is to bring H on a the form [tex]H = \left(\mathbf{A}\mathbf{s}\right)\cdot\left(\mathbf{B}\mathbf{S}\right)[/tex] where A and B are matrices describing rotations.

[tex]H=\left[\left(1-\gamma^{2}\right)\underbrace{\left(

\begin{array}{ccc}

1 & 0 & \theta \\

0 & 1 & 0 \\

-\theta & 0 & 1

\end{array} \right)}_{R_{y}(\theta)+O(\theta^{2})}

\mathbf{s}\right]\cdot\mathbf{S} \;\; \approx \;\; \left(1-\gamma^{2}\right)\left(R_{y}(\theta) \mathbf{s}

\right)\cdot \mathbf{S} \;, \qquad \theta=\frac{-2\gamma}{1-\gamma^{2}}

[/tex]

That is, H is a vector product between spin S and a spin s that has been rotated around the y-axis with angle theta.

In this case the matrix cannot be direct identified as a rotation around the x,y or z axis. The questions are now:

1) can it be a rotation around another axis?

2) Is it possible to write it as [tex]H = \left(\mathbf{A}\mathbf{s}\right)\cdot\left(\mathbf{B}\mathbf{S}\right)[/tex] where A and B are matrices describing rotations?

I would be glad if anybody has an idea about how to deal with this.

[tex]\mathbf{s}=\left(

\begin{array}{c}

s_{x} \\ s_{y} \\ s_{z}

\end{array} \right),\; \mathbf{S}=\left(

\begin{array}{c}

S_{x} \\ S_{y} \\ S_{z}

\end{array} \right)[/tex]

[tex]H=\left[\left(

\begin{array}{ccc}

1-\left(1-C_{z}^{2}+C_{x}^{2}\right)\gamma^{2}& 0 & -2\gamma\left(1-C_{x}C_{z}\gamma\right) \\

0 & 1-\left(1+C_{z}^{2}+C_{x}^{2}\right)\gamma^{2} & 0 \\

2\gamma\left(1-C_{x}C_{z}\gamma\right)& 0 &

1-\left(1+C_{z}^{2}-C_{x}^{2}\right)\gamma^{2}

\end{array} \right) \mathbf{s}\right]\cdot\mathbf{S}

[/tex]

(for the physics interested: this describes Kondo effect in a quantum dot with spin-orbit interaction)

The goal is to bring H on a the form [tex]H = \left(\mathbf{A}\mathbf{s}\right)\cdot\left(\mathbf{B}\mathbf{S}\right)[/tex] where A and B are matrices describing rotations.

For Cx=Cz=0 :

For Cx=Cz=0 :

[tex]H=\left[\left(1-\gamma^{2}\right)\underbrace{\left(

\begin{array}{ccc}

1 & 0 & \theta \\

0 & 1 & 0 \\

-\theta & 0 & 1

\end{array} \right)}_{R_{y}(\theta)+O(\theta^{2})}

\mathbf{s}\right]\cdot\mathbf{S} \;\; \approx \;\; \left(1-\gamma^{2}\right)\left(R_{y}(\theta) \mathbf{s}

\right)\cdot \mathbf{S} \;, \qquad \theta=\frac{-2\gamma}{1-\gamma^{2}}

[/tex]

That is, H is a vector product between spin S and a spin s that has been rotated around the y-axis with angle theta.

For Cx and Cz different from 0 :For Cx and Cz different from 0 :

In this case the matrix cannot be direct identified as a rotation around the x,y or z axis. The questions are now:

1) can it be a rotation around another axis?

2) Is it possible to write it as [tex]H = \left(\mathbf{A}\mathbf{s}\right)\cdot\left(\mathbf{B}\mathbf{S}\right)[/tex] where A and B are matrices describing rotations?

I would be glad if anybody has an idea about how to deal with this.