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Multiplication between vector and vector operator

  1. Jun 30, 2013 #1
    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.
     
  2. jcsd
  3. Jun 30, 2013 #2
    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).
     
  4. Jun 30, 2013 #3
    In quantum mechanics, both of those objects are operators. The operator [itex]\mathbf{r}[/itex] operates on state vectors in real space, whereas the operator [itex]\sigma[/itex] operates on spinors in spin space. You can treat (symbolically) the dot product between [itex]\mathbf{r}[/itex] and [itex]\sigma[/itex] as a normal dot product between two vectors, i.e.

    [tex]\mathbf{r}\cdot\sigma=x\sigma_x+y\sigma_y+z\sigma_z[/tex]

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

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

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