Chapter 6: Product operators - The James Keeler Group

6.5 Multiple quantum terms6.5.1 Coherence orderIn NMR the directly observable quantity is the transverse magnetization, whichin product operators is represented by terms such as I 1xand 2I 1zI 2 y. Such termsare examples of single quantum coherences, or more generally coherences withorder, p = ±1. Other product operators can also be classified according tocoherence order e.g. 2I 1zI 2 zhas p = 0 and 2I 1xI 2yhas both p = 0 (zero-quantumcoherence) and ±2 (double quantum coherence). Only single quantumcoherences are observable.In heteronuclear systems it is sometimes useful to classify operatorsaccording to their coherence orders with respect to each spin. So, for example,2I 1zI 2 yhas p = 0 for spin 1 and p = ±1 for spin 2.6.5.2 Raising and lowering operatorsThe classification of operators according to coherence order is best carried outbe re-expressing the Cartesian operators I xand I yin terms of the raising andlowering operators, I +and I –, respectively. These are defined as followsI = I + iI I = I − iI[6.6]+ x y − x ywhere i is the square root of –1. I +has coherence order +1 and I –has coherenceorder –1; coherence order is a signed quantity.Using the definitions of Eqn. [6.6] I xand I ycan be expressed in terms of theraising and lowering operators( ) = ( )11Ix = 2 I+ + I− Iy 2 i I+ – I−[6.7]from which it is seen that I xand I yare both mixtures of coherences with p = +1and –1.The operator product 2I 1xI 2xcan be expressed in terms of the raising andlowering operators in the following way (note that separate operators are usedfor each spin: I 1±and I 2±)112I1 I = 22 × ( 2 I + 1I )× ( 1 2 I + 2I )x x + − + 2−[6.8]11= 2 ( I1 +I2+ + I1 −I2− )+ 2 ( I1 +I2− + I1 −I2+)The first term on the right of Eqn. [6.8] has p = (+1+1) = 2 and the second termhas p = (–1–1) = –2; both are double quantum coherences. The third and fourthterms both have p = (+1–1) = 0 and are zero quantum coherences. The value ofp can be found simply by noting the number of raising and lowering operatorsin the product.The pure double quantum part of 2I 1 xI 2xis, from Eqn. [6.8],1double quantum part[ 2I 1 xI 2 x]= 2 ( I 1 +I 2 ++ I 1 −I2 − ) [6.9]The raising and lowering operators on the right of Eqn. [6.9] can be reexpressedin terms of the Cartesian operators:6–12