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Spinor spherical harmonics

Inquantum 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]

Properties[edit]

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

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

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 number and m is the total magnetic quantum number.

Under a parity operation, we have

${\displaystyle PY_{l,sj,m}=(-1)^{l}Y_{l,s,j,m}.}$

For spin-1/2 systems, they are given in matrix form by^[1][3][5]

${\displaystyle Y_{l,\pm {\frac {1}{2}},j,m}={\frac {1}{\sqrt {2{\bigl (}j\mp {\frac {1}{2}}{\bigr )}+1}}}{\begin{pmatrix}\pm {\sqrt {j\mp {\frac {1}{2}}\pm m+{\frac {1}{2}}}}Y_{l}^{m-{\frac {1}{2}}}\\{\sqrt {j\mp {\frac {1}{2}}\mp m+{\frac {1}{2}}}}Y_{l}^{m+{\frac {1}{2}}}\end{pmatrix}}.}$

where ${\displaystyle Y_{l}^{m}}$ are the usual spherical harmonics.

References[edit]

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