Recurrence and differential relations for spherical spinors

Abstract

We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) ${\Omega }_{\kappa \mu }(\mathbf{n})$ used in relativistic atomic, molecular, and solid state physics, as well as in relativistic quantum chemistry. First, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot \mathbf{B}{\Omega }_{\kappa \mu }(\mathbf{n})$ and {$\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C}){\Omega }_{\kappa \mu }(\mathbf{n})$}, where $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are either of the following vectors or vector operators: $\mathbf{n}=\mathbf{r}/r$ (the radial unit vector), ${\mathbf{e}}_0$, ${\mathbf{e}}_{\pm 1}$ (the spherical, or cyclic, versors), $\mathbf{\sigma }$ (the $2\times 2$ Pauli matrix vector), $\hat{\mathbf{L}}=-i\mathbf{r}\times \nabla I$ (the dimensionless orbital angular momentum operator; $I$ is the $2\times 2$ unit matrix), $\hat{\mathbf{J}}=\hat{\mathbf{L}}+1/2\mathbf{\sigma }$ (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot \mathbf{B}F(r){\Omega }_{\kappa \mu }(\mathbf{n})$ and $\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C})F(r){\Omega }_{\kappa \mu }(\mathbf{n})$, where at least one of the objects $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ is the nabla operator $\nabla$, while the remaining ones are chosen from the set $\mathbf{n}$, ${\mathbf{e}}_0$, ${\mathbf{e}}_{\pm 1}$, $\mathbf{\sigma }$, $\hat{\mathbf{L}}$, $\hat{\mathbf{J}}$.

LaTeX, 12 pages

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