- #1
Phileas.Fogg
- 32
- 0
Hello,
I'm studying the Heisenberg Model. Given the Hamiltonian
[tex]H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2[/tex]
with
[tex] \begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation} [/tex]
[tex] \sigma_{x,y,z} \quad {\text are \; the \; Pauli \; Matrices} [/tex]
Supposed, there are two electrons interacting antiferromagnetically.
So I have a spin-spin interaction. The first is up and the second spin is down.
The texts says that this results in the following Hamiltonian:
[tex] \begin{equation} H = - J \; \left(\begin{array}{cccc}
0,5 & 0 & 0 & 0 \\
0 & -0,5 & -1 & 0 \\
0 & -1 & -0,5 & 0 \\
0 & 0 & 0 & 0,5 \end{array} \right) \end{equation} [/tex]
I know it has something to do with the tensor product of both spin vector operators.
I tried to calculate by myself, but didn't get the correct result.
Could anyone explain, how this works?!
Thanks.
Regards,
Phileas
I'm studying the Heisenberg Model. Given the Hamiltonian
[tex]H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2[/tex]
with
[tex] \begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation} [/tex]
[tex] \sigma_{x,y,z} \quad {\text are \; the \; Pauli \; Matrices} [/tex]
Supposed, there are two electrons interacting antiferromagnetically.
So I have a spin-spin interaction. The first is up and the second spin is down.
The texts says that this results in the following Hamiltonian:
[tex] \begin{equation} H = - J \; \left(\begin{array}{cccc}
0,5 & 0 & 0 & 0 \\
0 & -0,5 & -1 & 0 \\
0 & -1 & -0,5 & 0 \\
0 & 0 & 0 & 0,5 \end{array} \right) \end{equation} [/tex]
I know it has something to do with the tensor product of both spin vector operators.
I tried to calculate by myself, but didn't get the correct result.
Could anyone explain, how this works?!
Thanks.
Regards,
Phileas