Operator formalisms: An overview

Abstract

ABSTRACT:Operator formalisms are mathematical recipes for simplifying the manipulations of the density matrix. Because the density matrix is a vital tool in describing and analyzing magnetic resonance experiments, many approaches have been developed. This article gives an overview of a number of different methods: spherical tensors, fictitious spin-1/2, single-transition operators, product operators, superspin methods, and others. In principle, they all must give the same answer, because an exact description of magnetic resonance phenomena is usually within reach. The choice of the formalism for the user, therefore, depends on various personal decisions. Among these decisions is the choice between spherical and Cartesian tensors, between Hilbert space and Liouville space, between commutators and matrix elements, and so on. We do not go into the details of any of the formalisms but rather try to compare their approaches at a fairly general level. The quadrupolar echo pulse sequence is used as an example of the application of the formalisms. The aim of this overview is to give readers enough of a picture so that they can make an intelligent choice for themselves.

Complete description of the interactions of a quadrupolar nucleus with a radiofrequency field. Implications for data fitting

“…Since the derivative is a function of the parameter itself for a non-linear problem, we can optimize our range of parameters. For instance, in the inversion-recovery experiment for measuring T 1 , the derivatives show that the choice of variable delays for recovery is not all that critical [67,68], as we already knew from experience.…”

“…The theory is exact and completely general, with no restrictions on the relative sizes of the RF, Zeeman and quadrupole interactions and it has proved useful in the case of third-order perturbations to the central transition [39]. Explicit formulae for the results can not be derived for the general case, but symbolic solutions are available for simple cases [40].…”

Challenges in numerical simulations of solid-state NMR experiments: Spin exchange pulse sequences

“…Second, in cases of chemical exchange, e.g. two-site jumps, the rate constant, k ij , for this exchange enters in addition to the longitudinal mixing [64,65] leading to off-diagonal elements of the exchange matrix given by…”

Simulating spin dynamics in NMR with a new computer program intended for education: Insensitive

“…Aside from the basic energy level diagram, these are the vector model, the product operator formalism (1)(2)(3), and the density matrix approach (3,4) of statistical quantum mechanics. While the latter two models provide a much deeper insight into and a more complete description of the spin dynamics, they operate on a much higher level of abstraction.…”