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 $|\chi\rangle$.