The resulting density matrix a3 is just the sum of the enhanced (by a factor of 4) and antiphase 13C single quantum coherence and the normal in-phase 13C SQC.


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 Bo field, the B i field, and with each other.

The symbols we have been using to represent spin states (Ix, Sy, 2Iy Sz, 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 Ia+ 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:

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