Nikiya Anton Bettey 2021: 09-28-2021-1813 - spinor spherical harmonics

Wednesday, September 29, 2021

09-28-2021-1813 - spinor spherical harmonics

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In quantum mechanics, the spinor spherical harmonics^[1] (also known as spin spherical harmonics,^[2] spinor harmonics^[3] and Pauli spinors^[4]) are special functions defined over the sphere. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential.^[3] The spinor spherical harmonics are sometimes called Pauli central field spinors, in honor to Wolfgang Pauli who employed them in the solution of the hydrogen atom with spin–orbit interaction.^[1]

roperties[edit]

The spinor spherical harmonics $Y l, s, j, m$ are the spinors eigenstates of the total angular momentum operator squared:

{\begin{aligned}\mathbf {j} ^{2}Y_{l,s,j,m}&=j(j+1)Y_{l,s,j,m}\\\mathrm {j} _{\mathrm {z} }Y_{l,s,j,m}&=mY_{l,s,j,m}\;;\;m=-j,-(j-1),\cdots ,j-1,j\\\mathbf {l} ^{2}Y_{l,s,j,m}&=l(l+1)Y_{l,s,j,m}\\\mathbf {s} ^{2}Y_{l,s,j,m}&=s(s+1)Y_{l,s,j,m}\end{aligned}}

where $j = l + s$ , where $j$ , $l$ , and $s$ are the (dimensionless) total, orbital and spin angular momentum operators, j is the total azimuthal quantum numberand m is the total magnetic quantum number.