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So the time course can be written as:

Since yBx t = ©, the pulse rotation in radians.

That is a lot of work to find out what we already knew, but it establishes a general method to figure out what happens to a spin state when it finds itself in an environment described by a particular Hamiltonian. Here are a few Hamiltonians for common time-dependent situations we have looked at

RF Pulse on y' axis Chemical shift evolution J-Coupling evolution (Av » J)

In each case, t times the multiplier in front of the Hamiltonian becomes the argument (in radians) of the sine and cosine functions when we write out the time course of the spin state.

10.8.3 The Ideal Isotropic Mixing (TOCSY Spin-Lock) Hamiltonian

Now let's look at something we do not know the answer to: the ideal isotropic mixing Hamiltonian. This is the ideal TOCSY mixing sequence that leads to in-phase to in-phase coherence transfer. The ideal sequence of pulses creates this average environment expressed by the Hamiltonian. The "Zeeman" Hamiltonian that represents the chemical shifts goes away and we have only the isotropic (i.e., same in all directions) /-coupling Hamiltonian:

We already put together the matrix representation of HJ :

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