- = The spin-orbit operator of a single electron has the form of a symbolic scalar product of two vectors, (, s), where = (x, y, z), = ($x, y, z) are the Cartesian components of the orbital angular momentum and the spin. 1. Using the matrix representation of the spin operator, derive a compact matrix representation of the spin-orbit operator. For example, the z t "matrix" will have the form 0 (:) 0 X Ym 0 2. Find the result of this operator's action on a "spin-up" ( and a "spin-down" ) function. You will 0 Yim find the exact expressions of familiar relations useful: (x + iy)Ym VI(1 + 1) m(m + 1)71, m+1 (v iy)Ym VI(1 + 1) mam 1)Y, m1 = = -Y10 .) 3. Verify that a two-component function is an eigenfunction of the spin-orbit operator. What does it tell V2Yu about the structure of wavefunctions that describe electrons in the presence of spin-orbit interaction? 11