Product Operator Formalism: A Brief Introduction

Multiple spin systems allow for the presence of quantum coherence and spin-spin coupling interactions. Traditionally, density operator formalism is used to describe these systems; however, the use of this formalism often clouds intuition making it a cumbersome tool. Here we introduce a third formalism constructed from the density operator formalism with a direct connection to a non-classical intuitive vector-like representation. After introducing the method, a simple example of its use is given when describing the COSY 2D-NMR experiment. Introduction When describing spin systems generally two approaches are used: the classical vector formalism [1] or the density operator formalism [2]. Here we seek to briefly introduce the product operator formalism developed by Sørensen [3], as an alternative to the standard approaches. In Sect. 1 and 2, we describe the classical vector and density operator formalisms. In Sect. 3 we introduce the product operators for a two spin system. In Sect. 4 we describe the derivation of the product operator formalism from the density operator formalism. In Sect. 5 we introduce the nonclassical vector-like interpretation of the product operator formalism. In the final section we present a working example of the product operator formalism applied to the Correlation Spectroscopy (COSY) NMR experimental technique. Classical Vector Formalism The classical vector approach is rooted in the idea of equating a classically precessing charged particle in a magnetic field with that of a particle with some spin, all interactions between the ‘spinning’ particle and any external interactions (outside of the field the particle is ‘spinning’ in) are described by a torque being exerted on the particle. Extending on these same principles, the Bloch equations can also be derived (see Mirau [1]) to directly connect the classical model with observable changes in magnetization. A problem occurs at this point in that the inclusion of a second spin to allow for spin-spin interactions is not a direct extension, and worse will not be able to account for coherence between different spin systems. Density Operator Formalism Density operator formalism, in contrast to the classical vector formalism, is rooted in quantum mechanics. The density operator is defined by: