Spins raising operator

The spin raising operator (see Eq. (4.3)) may now be applied to this result, and we obtain... 

Now consider the commutator of J3 with the isotopic spin raising operator T+ = Ti -I- iT2 which, in the Condon-Shortley phase convention (Condon and Shortley, 1963) has the following effect ... 

Just like the electron spin-raising and spin-lowering operators defined in Equations 7.15 and 7.16, we have the analogous operators for the nuclear spin I ... 

Analogous to (1.169), the electron-spin raising and lowering operators are... 

Also, since S + is a raising operator, we must have / — ca, where c is some constant. We can evaluate c by use of the normalization of the spin functions and the Hermitian property of Sx and Sy one finds (Problem 1.8) c = h. Choosing the phase of c as zero, we have... 

The raising operator I+, acting as an operator, raises the a state of a single spin to the p state. All of these operators can be represented as matrices. In the case of the homonuclear two-spin system (Ha and Hb), these are 4 x 4 matrices. For example, the raising operator 1+ can be represented by the following matrix, which acts on the vector that describes the aa state to give a new vector that describes the Pa state ... 

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 + =... 

Since a and p are often referred to as spin up ( ) and spin down ( ) states, and S" are called spin raising and lowering operators, respectively. Looking at Eqs. (17.4) one gets the impression that there is some analogy between creation/ annihilation operators and the spin operators S /S . To work out the connection between spin operators and second quantization, one has to analyze some points in more detail. The spin operators obey the following commutation rules ... 

In this formula, we identify four contributions The first term is formally identical to the ROHE expression also found in the DODS case. However, care must be taken that it is actually different, because the numbers of a- and /3-electrons are not good quantum numbers in the GCHF case. The second term is the z-noncollinearity contribution. The third term is formally analogous to the spin contamination of a DODS wave function as defined in [7, 8]. Finally, the last term is the square of the expectation value of the lowering or raising operator ... 

As a first step in the study of the Heisenberg ferromagnet, we express the dot product of two spin operators, with the help of the spin raising 5+ and lowering 5 operators (for details see Appendix B), as... 

Moreover, we can determine that the total spin value of this new state is 5 = 0, because applying the lowering or the raising operator on it gives 0. This completes the analysis of the possible spin states obtained by combining two spin-1/2 particles which are the following ... 

The raising and lowering operators for spin angular momentum as defined by equations (5.18) are... 

Our next goal is to transform this expression into one based on the total electron spin operator, S = si + s2. The first three terms can be simplified by making use of the identity (derived using raising and lowering operators) ... 

If we expand the hyperfine term of the spin Hamiltonian and write the operators in terms of raising and lowering operators ... 

The standard procedure used to solve Eq. (29) is to transform the spin operators into fermionic operators [61]. Let us define the raising and lowering operators... 

A description in terms of local spins, however, raises the question of how to define these localized surrogate spins, where we have to move from one-electron spin operators Sj to multi-electron spin operators Sa that are located at one center and summarize all electronic contributions attributed to this center. This can be solved by the introduction of projection operators Pa (33,113,114,122-124). The projection operators do not alter any property of the molecule but divide it into local basins A. They add up to the identity operator,... 

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