Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\ \sqrt{l \mp m +\frac{1}{2}}Y_{l, m + \frac{1}{2}} \end{pmatrix}. $$ How to prove the relation $$ (\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -r\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}? $$ I tried to rewrite the $(\hat {\sigma} \cdot \mathbf r )$ as $$ (\hat {\sigma} \cdot \mathbf r ) = r\begin{pmatrix} \cos(\theta ) & \sin(\theta )\,e^{-i\varphi } \\ \sin(\theta )\, e^{i \varphi } & -\cos(\theta ) \end{pmatrix}, $$ but it's hard to use recurrence relations for associated Legendre functions after that. Is there some hint which simplifies the proof?

Maybe, I can first show that $(\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -ra\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}$, because the parity of the left side is opposite to the parity of $\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}}$, and $(\hat {\sigma} \cdot \mathbf r )$ commutes with rotation operator, because refers to the scalar pdoduct?


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