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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

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# Interacting Spins , Heisenberg Hamiltonian , Tensor product

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