Because I and S operators commute, a nonselective pulse can be
represented by a pulse onIfirst, followed by a pulse onS(or by a pulse
acting onSfollowed by a pulse acting onI),
^1 !
Ix
^2 )
Sx
^3 : ½ 2 : 241
Rotations of products. The effect of a pulse applied selectively to the
Sspin of a product term such as 2IxSzis obtained using the rule that
rotations only affect operators of the same spin. In other words, theIx
part of the product operator remains untouched by the pulse to the
Sspin and theSzterm is rotated normally. The result obtained is
2 IxSz )
Sx
2 IxSzcos 2 IxSysin : ½ 2 : 242
Rotations involving the same operator. Operators are unaffected by
rotations about themselves because an operator and the exponential of
an operator commute. For example,
Ix )
Ix
Ix, ½ 2 : 243 2 IySz )(^) SSzt
2 IySz: ½ 2 : 244
2.7.4 SINGLE-QUANTUMCOHERENCE AND
OBSERVABLEOPERATORS
The single-quantum coherence termIxcan be expressed using [2.210]
and [2.218] as
Ix¼^12 ðIþS^ þIS^ Þþ^12 ðIþS^ þIS^ Þ: ½ 2 : 245
This operator, involving a transverse Cartesian component, results from
the sum of the single-quantum transitions of theIspin. Evolution under
the free-precession Hamiltonian yields
expfiHtgIxexpfiHtg¼^12 ðIþS^ exp½ið (^) IþJÞt
þIS^ exp½ið (^) IþJÞtÞ
þ 21 ðIþS^ exp½ið (^) IJÞt
þIS^ exp½ið (^) IJÞtÞ: ½ 2 : 246
88 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY