Ladder operators for electron spin

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 (5.114) and (5.115), the raising and lowering operators for spin angular momentum are 

We now use the Hermitian property of and Sy. For an operator A that acts on functions of the continuous variable x, the Hermitian property is 

Taking the complex conjugate of this equation and using (10.60) and (10.62), we have 

Nuclei with 1 = 0 (for example, 0) don t split proton NMR peaks. It turns out that nuclei with / j (for example, Cl, Cl, N) generally don t split proton NMR peaks. The nucleus has / = j and does split proton NMR peaks. Also, a qnantum-mechanical analysis shows that the spin-spin interactions between equivalent protons don t affect the NMR spectrum. 

NMR spectroscopy is the premier strnctnral research tool in organic chemistry, and special NMR techniques allow the structures of small proteins to be determined with the aid of NMR. 

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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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