The Hamiltonian Matrix Strong Coupling and Ideal Isotropic TOCSY Mixing

8 THE HAMILTONIAN MATRIX STRONG COUPLING AND IDEAL ISOTROPIC (TOCSY) MIXING 

I have done everything I can to avoid using the H word, but now that we have learned how to represent product operators in a matrix form it is a short step to working with the Hamiltonian. The Hamiltonian is a representation of the environment the spins find themselves in—it contains the energies of all of the interactions of spins with the B0 field, the B1 field, and with each other. 

The symbols we have been using to represent spin states (I, S-y, 2IySz, etc.) of the entire ensemble of spins are actually operators they can operate on a spin state (of a single spin pair in our Ha, Hb system) and spit out another spin state. We already saw this with the raising and lowering operators  

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  

These vectors are just a column of numbers representing the coefficients of the four pure spin states ci, C2, C3 and c4. They do not describe the whole ensemble of spins, just one Ha - Hb pair. 

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