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Quantum Simulation Entanglement Theory Quantum Correlations Quantum Chemistry Quantum State Preparation Representation

Recurrence and differential relations for spherical spinors

arXiv

Authors: Radosław Szmytkowski

Year

2010

Paper ID

10418

Status

Preprint

Abstract Read

~2 min

Abstract Words

222

Citations

N/A

Abstract

We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) Ω_κμ$mathbf{n}$ 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}cdotmathbf{B} Ω_κμ$mathbf{n}$ and {mathbf{A}cdot$mathbf{B}timesmathbf{C}$ Ω_κμ$mathbf{n}$}, 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}_pm1 (the spherical, or cyclic, versors), boldsymbolσ the $2times2$ Pauli matrix vector, hat{mathbf{L}}=-imathbf{r}timesboldsymbol{nabla}I the dimensionless orbital angular momentum operator; $I$ is the $2times2$ unit matrix, hat{mathbf{J}}=hat{mathbf{L}}+1/2boldsymbolσ (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the expressions mathbf{A}cdotmathbf{B} F(r)Ω_κμ$mathbf{n}$ and mathbf{A}cdot$mathbf{B}timesmathbf{C}$ F(r)Ω_κμ$mathbf{n}$, where at least one of the objects mathbf{A}, mathbf{B}, mathbf{C} is the nabla operator boldsymbol{nabla}, while the remaining ones are chosen from the set mathbf{n}, mathbf{e}₀, mathbf{e}_pm1, boldsymbolσ, hat{mathbf{L}}, hat{mathbf{J}}.

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We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) Ωκμ(mathbfn) used in...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:1011.3433 [2] arXiv https://arxiv.org/abs/1011.3433 [3] Publisher https://arxiv.org/abs/1011.3433

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