Proof of Dot and Cross Product of Arbitrary Vectors with Pauli Spin Operator

[indigojoker]

I need to show:

$$(\mathbf{\sigma} \cdot \mathbf{a})(\mathbf{\sigma} \cdot \mathbf{b})=\mathbf{a} \cdot \mathbf{b} I + i \mathbf{\sigma} \cdot (\mathbf{a} \times \mathbf{b})$$

where a and b are arbitrary vectors, sigma is the pauli spin operator.

I was just wondering what the dot product and cross product were. Because a and b can be 2x1, 2x2, 2x3, etc... I'm not sure how to take a dot product of matricies much less a cross product. Since it specifies dot and cross, i assume that it is not just a regular matrix mulitpilication, however, i do not know how to take the dot and cross product of matrices. Any suggestions would be appreciated.

 

[nrqed]

I need to show:

$$(\mathbf{\sigma} \cdot \mathbf{a})(\mathbf{\sigma} \cdot \mathbf{b})=\mathbf{a} \cdot \mathbf{b} I + i \mathbf{\sigma} \cdot (\mathbf{a} \times \mathbf{b})$$

where a and b are arbitrary vectors, sigma is the pauli spin operator.

I was just wondering what the dot product and cross product were. Because a and b can be 2x1, 2x2, 2x3, etc... I'm not sure how to take a dot product of matricies much less a cross product. Since it specifies dot and cross, i assume that it is not just a regular matrix mulitpilication, however, i do not know how to take the dot and cross product of matrices. Any suggestions would be appreciated.

Just think of the ${\vec \sigma}$ as vectors. So, for example, ${\vec a} \cdot {\vec \sigma}$ is simply $a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3$ and so on. So you treat the ${\vec \sigma}$ as vectors while being careful to rememember that the components do not coommute. And then , you may write them as matrices to complete your calculation.

 

[indigojoker]

thanks nrqed. are vectors a and b 2x1? or is it a general 2xn?

Or is a=x1+x2+x3, where x1, x2, x3 are 2x1 vectors? I'm not too sure how general this should be