(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm supposed to derive the following:

[tex] \left({\bf A} \cdot {\bf \sigma} \right) \left({\bf B }\cdot {\bf \sigma} \right) = {\bf A} \cdot {\bf B} I + i \left( {\bf A } \times {\bf B} \right) \cdot {\bf \sigma} [/tex]

using just the two following facts:

Any 2x2 matrix can be written in a basis of spin matrices:

[tex] M = \sum{m_\alpha \sigma_\alpha} [/tex]

which means that the beta-th component is given by

[tex] m_\beta = \frac{1}{2}Tr(M \sigma_\beta) [/tex]

2. Relevant equations

listed above...

3. The attempt at a solution

It should just be a left side= right side proof.

I started by saying [tex] \left({\bf A} \cdot {\bf \sigma} \right) \left({\bf B }\cdot {\bf \sigma} \right) =

\left(\sum_\alpha a_\alpha \sigma_\alpha \cdot \sigma \right) \left(\sum_\alpha b_\alpha \sigma_\alpha \cdot \sigma \right) =

\left( \sum_\beta a_\alpha \delta_{\alpha \beta} \right) \left( \sum_\gamma b_\alpha \delta_{\alpha \gamma} \right) =

\sum_\beta a_\beta \sum_\gamma b_\gamma =

\frac{1}{2} Tr(A \sigma_\beta) \frac{1}{2} Tr(B \sigma_\gamma) [/tex]

Not sure if this is even the right way to start, and I can't see at all where I would go from here to get the appropriate RHS of the identity I'm proving. Any ideas?

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# Homework Help: Deriving a vector identity using Pauli spin matrices

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