Spin spherical harmonics

In quantum mechanics, spin spherical harmonics $Y l, s, j, m$ are 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}\end{aligned}}

where $j = l + s$ . They are the natural spinorial analog of vector spherical harmonics.

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

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

Spin spherical harmonics are used in analytical solutions to Dirac equation in a radial potential.

Notes

Biedenharn, L. C.; Louck, J. D. (1981), Angular momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics, 8, Reading: Addison-Wesley, p. 283, ISBN 0-201-13507-8

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