Coupled Spins: Density Matrix and Product Operator Formalism

Abstract

In the previous chapter the coupling between two spins was analyzed using the wavefunctions associated with each of the quantum states. This type of analysis is quite satisfactory for determining the appearance of the one-dimensional NMR spectrum of the coupled spins. However, this approach is completely impractical for calculations on an ensemble of spins that are subject to multiple excitation pulses in typical NMR experiments. Consequently, we return to the density matrix to analyze the evolution of the coupled spins. In this analysis we will only assume weak coupling, of the AX kind. This assumption is not a severe restriction since heteronuclear couplings, such as between protons and carbon or carbon and nitrogen, are those that are used for magnetization transfer by various NMR experiments.

This chapter begins with the analysis of a one-pulse experiment using the density matrix. The analysis follows the same treatment as given in Chapter 6 for an uncoupled spin. The density matrix will then be represented by product operators and a series of rules that describe the transformation of the product operators by pulses and free evolution will be introduced. These rules will greatly simplify the analysis of complicated pulse sequences.