Ladder operators, spin orbital

PROBLEM 3.5.6. One can define linear ladder operators for angular momentum (orbital or spin) the raising operator + = Lx + iLy and the lowering operator =LX — iLy. (a) Verify that brute-force expansion yields + =... 

Every primitive ket is an eigenfunction of the S. operator, a result of the direct product form of the spin-orbital basis. The ladder operators S+ and S defined as... 

In determining the preferred spin orientation, the most important interaction between occupied and unoccupied spin levels is the one with the smallest energy gap Ae = (e,—Cj). The d-levels of same m values (e.g., between xz and yz, ancj between xy and x —y) interact through the operator z to give nonzero (/ Hsoli) ( Hso i). The d-levels of different m values with Am= i (e.g., between xz/yz and xy/(x — y ), and between z and xz/yz) interact through the ladder operators L+ and to give nonzero (/ IHjoIj) and (/ n soli). The z orbital cannot interact with the xy/ (x —y ) orbitals under SOC because their m values differ by Am = 2. 

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