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NMR Product-Operator Calculations in Mathematica

J. Shriver
Journal / Anthology

Journal of Magnetic Resonance
Year: 1991
Volume: 94
Page range: 612-616

It has become standard practice to describe many multiple-pulse NMR experiments by using the product-operator formalism. The formalism is a rigorous analytical tool which allows for an intuitive understanding of the effect of pulses in modern NMR experiments, many of which cannot be adequately described by using the classical vector model. The density oeprator is represented as a linear combination of product operators, or coherences, each of which can be visualized by using vector diagrams. Only a few of the coherences can be directly observed experimentally, but the formalism allows for the visualization of the evolution of a spin system during a multiple-pulse experiment in terms of the relative importance of the operators in the expansion, thereby facilitating the understanding of such techniques as polarization transfer and multiple-qualnum filtration.


We point out here the ease with which a set of procedures and rules may be incorporated into Mathematica to allow product-operator calculations to be performed interactively by computer. Mathematica is a new computer language which contains many of the advantages of procedural languages such as C with the additional advantage of having an interpreter. It is especially useful for performing symbolic computations and can be easily extended by incorporation of user-defined algebraic rules. The rule-based programming capability of Mathematica makes it very useful for manipulation and especially algebraic simplification of product-operator expressions. In addition, the language allows for the creation of procedures to describe the effect of pulses, precession, and J coupling. All of these have been packaged as a Mathematica notebook for running on a Macintosh computer.

*Science > Physics > Quantum Physics