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In this lecture, I give a short and essential overview of some important approximation theories and mathematical tricks used in spin dynamics to cope with otherwise intractable problems. I discuss perturbation theory, average Hamiltonian theory and interaction frame transformations. Perturbation theory is used to find approximate solutions to complex or insoluble eigenvalue problems when the Hamiltonian can be divided into a large, unperturbed, and a small, perturbation, term. Average Hamiltonian theory provides an approximate solution to the Liouville–von Neumann equation when the Hamiltonian is time-dependent. An interaction frame transformation is a recipe for deriving the form that Hamiltonians assume when the reference frame is transformed by rotations.

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