Introduction to average Hamiltonian theory. I. Basics

Understanding the dynamics of electron or nuclear spins during a magnetic resonance experiment requires to solve the Schrödinger equation for the spin system considering all contributions to the Hamiltonian from interactions of the spins with each other and their surroundings. In general, this is a difficult task as these interaction terms can be both time-dependent and might not commute with each other. A powerful tool to analytically approximate the time evolution is average Hamiltonian theory, in which a time-independent effective Hamiltonian is taking the place of the time-dependent Hamiltonian. The effective Hamiltonian is subjected to the Magnus expansion, allowing to calculate the effective Hamiltonian to a certain order. The goal of this paper is to introduce average Hamiltonian theory in a rigorous but educational manner. The application to two composite pulses in NMR spectroscopy is used to demonstrate important aspects of average Hamiltonian theory.