Harcourt's theorem: Difference between revisions

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In [[NMR spectroscopy]], the '''product operator formalism''' is a method used to determine the outcome of [[pulse sequence]]s in a rigorous but straightforward way. With this method it is possible to predict how the bulk [[magnetization]] evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism.
 
In this model, for a single [[spin (physics)|spin]], four base operators exist: <math>I_x</math>, <math>I_y</math>, <math>I_z</math> and <math>E/2</math> which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for [[Quantum coupling|coupled systems]]. Using this approach, the evolution of the magnetization under free precession is represented by <math>I_z</math> and corresponds to a rotation about the z-axis with a [[phase angle]] proportional to the [[chemical shift]] of the spin in question:
 
<math>I_x \xrightarrow{\omega \tau I_z} \cos (\omega \tau)I_x - \sin (\omega \tau)I_y </math>
 
Pulses about the x and y axis can be represented by <math>I_x</math> and <math>I_y</math> respectively; these allow to interconvert the magnetization between planes and ultimately to observe it at the end of a sequence. Since every spin will evolve differently depending on its shift, with this formalism it is possible to calculate exactly where the magnetization will end up and hence devise pulse sequences to measure the desired signal while excluding others.
 
The product operator formalism is particularly useful in describing [[Two-dimensional nuclear magnetic resonance spectroscopy|experiments in two-dimensions]] like COSY, HSQC and HMBC.  
 
== References ==
*{{cite web | publisher = [[University of California, Irvine]] | title = Understanding NMR Spectroscopy | author = James Keeler | format = reprinted at [[University of Cambridge]] | url = http://www-keeler.ch.cam.ac.uk/lectures/Irvine/ | accessdate = 2012-08-05}}
*{{cite web|publisher=[[University of Texas]]|title=A Pictorial Representation of Product Operator Formalism|author=David Donne|coauthor=David Gorenstein|url=http://www.biophysics.org/Portals/1/PDFs/Education/donne.pdf|accessdate=2012-08-05}}
 
[[Category:Nuclear magnetic resonance]]

Latest revision as of 15:50, 14 November 2014

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