While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin).
These functions are used in analytical solutions to Dirac equation in a radial potential.
[3] The spinor spherical harmonics are sometimes called Pauli central field spinors, in honor to Wolfgang Pauli who employed them in the solution of the hydrogen atom with spin–orbit interaction.
[1] The spinor spherical harmonics Yl, s, j, m are the spinors eigenstates of the total angular momentum operator squared: where j = l + s, where j, l, and s are the (dimensionless) total, orbital and spin angular momentum operators, j is the total azimuthal quantum number and m is the total magnetic quantum number.
Under a parity operation, we have For spin-1/2 systems, they are given in matrix form by[1][3][5] where