The Product Operator Formalism Introduction

It is very difficult to describe coherence transfer using the vector model. To understand it we will need to expand our theoretical picture to include product operators. Product operators are a shorthand notation that describes the spin state of a population of spins by dividing it into symbolic components called operators. You might wonder why you would trade in a nice pictorial system for a bunch of equations and symbols. The best reason I can give is that the vector model is useless for describing most of the interesting NMR experiments, and product operators offer a bridge between the familiar vectors and the more formal and 

Using the vector model, when we want to describe the spin state of a particular nucleus, we can draw a vector in three-dimensional space, or we can describe the projection of that vector onto the three axes (the components of the vector). For example, a vector of length M0 on the — x axis could be described as 

If there is only one NMR line in the spectrum (a population of identical nuclei) there are only three product operators, and they correspond to the three components of the net magnetization vector. A complete description of a population of spins can be given by the spin state a  

These simple product operators precess in the x -y plane of the rotating frame at a frequency corresponding to the chemical shift in hertz relative to the center of the spectral window (the resonance offset Av = v0 — vr). The chemical shift frequency Av can also be represented as the angular velocity 2 in units of rad/s ( 2 = 2ttAv). Using 2 allows us to skip all the 2tt terms. 

Any complicated representation of a spin system in terms of I, Iy, and Iz can be described after a delay r by substituting the corresponding expression on the right side every time one of the terms on the left side (I, Iy, or Iz) occurs in the representation. For example, if we start with the spin state 

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