What happen in calculate $ (\boldsymbol{\sigma} \cdot \boldsymbol{J}) (\boldsymbol{\sigma} \cdot \boldsymbol{J} - 1) $?

Question

If I use $ \boldsymbol{J} = \boldsymbol{L} + \boldsymbol{S} = \boldsymbol{L} + \frac{\boldsymbol{\sigma}}{2} $, then \begin{align*} (\boldsymbol{\sigma} \cdot \boldsymbol{J}) (\boldsymbol{\sigma} \cdot \boldsymbol{J} - 1) & = (\boldsymbol{\sigma} \cdot \boldsymbol{L} + \frac{3}{2}) (\boldsymbol{\sigma} \cdot \boldsymbol{L} + \frac{3}{2} - 1) \\ & = (\boldsymbol{\sigma} \cdot \boldsymbol{L})^{2} + 2 \boldsymbol{\sigma} \cdot \boldsymbol{L} + \frac{3}{4} \\ & = L^{2} + \boldsymbol{\sigma} \cdot \boldsymbol{L} + \frac{3}{4} \\ & = L^{2} + 2 \boldsymbol{S} \cdot \boldsymbol{L} + S^{2} \\ & = (\boldsymbol{L} + \boldsymbol{S})^{2} \\ & = J^{2} \end{align*} where \begin{align*} (\boldsymbol{\sigma} \cdot \boldsymbol{L})^{2} = L^{2} + i\boldsymbol{\sigma} \cdot (\boldsymbol{L} \times \boldsymbol{L}) = L^{2} - \boldsymbol{\sigma} \cdot \boldsymbol{L} \end{align*} but if I use \begin{align*} (\boldsymbol{\sigma} \cdot \boldsymbol{J})^{2} = J^{2} - \boldsymbol{\sigma} \cdot \boldsymbol{J} \end{align*} then I have \begin{align*} (\boldsymbol{\sigma} \cdot \boldsymbol{J}) (\boldsymbol{\sigma} \cdot \boldsymbol{J} - 1) & = (\boldsymbol{\sigma} \cdot \boldsymbol{J})^{2} - \boldsymbol{\sigma} \cdot \boldsymbol{J} \\ & = J^{2} - 2\boldsymbol{\sigma} \cdot \boldsymbol{J} \\ & \neq J^{2} \end{align*} So which equation is right, and what wrong with another equation?

Thanks by LiZ.

Answer 1

Your first expression is correct, the second one is wrong. It is true that $$ (\vec{\sigma}\cdot\vec{L})^2=L^2-\vec{\sigma}\cdot\vec{L} \ , $$ but you cannot generalize this to $\vec{J}$: $$ (\vec{\sigma}\cdot\vec{J})^2\neq J^2-\vec{\sigma}\cdot\vec{J} \ , $$ because $\vec{S}=\vec{\sigma}/2$ and $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k\neq0$, so $ (\vec{\sigma}\cdot\vec{J})^2\neq\sigma_i\sigma_kJ_iJ_k$.

Answer 2

Your calculations are inconsistent because you need to rather assume in a simultaneous eigenspace of $J^2,S^2,L^2$ (which commute): $$ J^2 =j(j+1) \quad L^2=l(l+1) \quad S^2 = s(s+1) $$ From there, you can compute spin orbit coupling: $$ L\cdot S = \frac12(J^2-L^2-S^2) = \frac12[j(j+1)-l(l+1)-s(s+1)] $$ as well as: $$ S\cdot J = S^2+S\cdot J = \frac12[j(j+1)-l(l+1)+s(s+1)] $$ so your quantity is (both you results were wrong): $$ (\sigma\cdot J)[(\sigma\cdot J)-1] = [j(j+1)-l(l+1)+s(s+1)][j(j+1)-l(l+1)+s(s+1)-1] $$