Spin spherical harmonics

In quantum mechanics, spin spherical harmonics Y_{l, s, j, m} are spinors eigenstates of the total angular momentum operator squared:

$\begin{align} \mathbf j^2 Y_{l, s, j, m} &= j (j + 1) Y_{l, s, j, m} \\ \mathrm j_{\mathrm z} Y_{l, s, j, m} &= m Y_{l, s, j, m} \end{align}$

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

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

References

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