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}}

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

Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 978-0-691-07912-7

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