Spin spherical harmonics

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

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ 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} }[/math]

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

References

• Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 978-0-691-07912-7, https://archive.org/details/angularmomentumi0000edmo