# Spin spherical harmonics

Not to be confused with Spin-weighted spherical harmonics.

In quantum mechanics, spin spherical harmonics are spinors that are eigenstates of the square of the angular momentum operator, and so are the natural spinorial analog of vector spherical harmonics.

They are given in matrix form by[1]

$Y_{(j-\frac{1}{2},\frac{1}{2})jm}=\left(\begin{array}{c} \sqrt{\frac{j+m}{2j}}Y_{j-\frac{1}{2},m-\frac{1}{2}}\\ \sqrt{\frac{j-m}{2j}}Y_{j-\frac{1}{2},m+\frac{1}{2}}\end{array}\right)$

$Y_{(j+\frac{1}{2},\frac{1}{2})jm}=\left(\begin{array}{c} -\sqrt{\frac{j-m+1}{2j+2}}Y_{j+\frac{1}{2},m-\frac{1}{2}}\\ \sqrt{\frac{j+m+1}{2j+2}}Y_{j+\frac{1}{2},m+\frac{1}{2}}\end{array}\right)$

## Notes

1. ^ 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