Multiplication between vector and vector operator

[LagrangeEuler]

How this is defined?

$\vec{r}\cdot \vec{\sigma}$?
where $\vec{r}=(x,y,z)$ and $\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$. $\sigma_i$ are Pauli spin matrices.

 

[wotanub]

It's a 2x2 matrix... It seems you've stumbled on the way to represent the location of a point in space as a 2x2 matrix instead of a vector. I'm fuzzy on the rigorous details, but it has something to do with the relationship between SO(3) and SU(2).

 

[conana]

In quantum mechanics, both of those objects are operators. The operator \mathbf{r} operates on state vectors in real space, whereas the operator \sigma operates on spinors in spin space. You can treat (symbolically) the dot product between \mathbf{r} and \sigma as a normal dot product between two vectors, i.e.

\mathbf{r}\cdot\sigma=x\sigma_x+y\sigma_y+z\sigma_z

where the x,y,z,\sigma_x,\sigma_y,\sigma_z are all operators. Say we have a state |\psi\rangle=|\phi\rangle\otimes|\chi\rangle where |\phi\rangle is a state vector in real space and |\chi\rangle is a spinor. Then

\mathbf{r}\cdot\sigma|\psi\rangle=x\sigma_x|\psi\rangle+y\sigma_y|\psi \rangle+z\sigma_z|\psi\rangle
=x|\phi\rangle\otimes\sigma_x|\chi\rangle+y|\phi\rangle\otimes \sigma_y|\chi\rangle+z|\phi\rangle\otimes\sigma_z|\chi\rangle

where the operators x,y,z operate on the real space state vector |\phi\rangle and the operators \sigma_x,\sigma_y,\sigma_z operate on the spinor |\chi\rangle.