Abstract
Multiple‐pulse NMR experiments are a powerful tool for the investigation of molecules with coupled nuclear spins. The product operator formalism provides a way to understand the quantum evolution of an ensemble of weakly coupled spins in such experiments using some of the more intuitive concepts of classical physics and semi‐classical vector representations. In this paper I present a new way in which to interpret the quantum evolution of an ensemble of spins. I recast the quantum problem in terms of mixtures of pure states of two spins whose expectation values evolve identically to those of classical moments. Pictorial representations of these classically evolving states provide a way to calculate the time evolution of ensembles of weakly coupled spins without the full machinery of quantum mechanics, offering insight to anyone who understands precession of magnetic moments in magnetic fields.
Highlights
The past several decades have seen an explosion in the development of new and powerful experimental techniques investigating coupled nuclei using multiple-pulse NMR experiments. The sophistication of these techniques presents a pedagogical problem: How can these techniques be introduced to students and practitioners who may never have the opportunity to develop a deep understanding of the quantum mechanical formalism that provides a quantitative description of all aspects of the dynamics of coupled spins? In this paper I identify classically evolving states in a fully quantum mechanical treatment of the time evolution of coupled spins in multiple-pulse NMR experiments
In some systems the classical-quantum parallels are qualitative; in the time evolution of the quantum double-arrow states of coupled spins that I have discussed in this paper, the FIGURE 1 1 Representation of the COSY time evolution of the mixture corresponding to the I1z term in the product operator expansion [Color figure can be viewed at wileyonlinelibrary.com]
The existence of a classical analog for ensembles of weakly coupled spins can help guide the intuition of the NMR expert sophisticated in the use of quantum mechanics
Summary
The past several decades have seen an explosion in the development of new and powerful experimental techniques investigating coupled nuclei using multiple-pulse NMR experiments. In this paper I identify classically evolving states in a fully quantum mechanical treatment of the time evolution of coupled spins in multiple-pulse NMR experiments Pictorial representations of these quantum states have the potential to provide physical insight to anyone who understands classical precession of moments in magnetic fields. The arrows in this figure represent the two directions for which the simultaneous measurement of the component of angular momentum for each single spin are both certain to yield +ħ/2. These double-arrow states can be written in terms of the standard basis of projections along the z-axis. Two-spin states can be written jhA; /A; hB; /Bi jhA; /AiA jhB; /BiB cos hA 2
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