Ladder operators, spin

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

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 + =... 

Fig. 13a-d The number operators a N and b, and the ladder operators c and d F[ in the manifold of Floquet states nk) with n) the Fourier states corresponding to the spinning frequency ujr and fc) the Fourier states corresponding to the characteristic RF frequency ujc... 

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... 

The spin angular-momentum operators obey the general angular-momentum commutation relations of Section 5.4, and it is often helpful to use spin-angular-momentum ladder operators. 

From this relation it is evident that the components of the total spin and the ladder operators are analogous sums of one-electron operators... 

The ladder operators 5= = = S c iSy change the ms quantum number of the spin functions by the following action... 

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. 

One of these models is the spin- ladder with competing interactions of the ferro- and antiferromagnetic types at the F-AF transition line. The exact singlet ground-state wave function on this line is found in the special form expressed in terms of auxiliary Bose-operators. The spin correlators in the singlet state show double-spiral ordering with the period of spirals equal to the system size. 

One can take advantage of any spin or spatial symmetry in the Hamiltonian by symmetry adapting the metric matrices and thereby reducing the size of the 2-RDM to be optimized [16]. For the ladder model, we transform the RDMs to bonding and antibonding spaces and then Fourier transform to take advantage of the translational symmetry. We consider linear combination of creation and annihilation operators to form two disjoint one-electron subspaces... 

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