where is the resonance offset, is the phase of an applied rf pulse, and
! 1 ¼ B 1 , and B 1 is the rf field strength ( 13 ). This geometrical
interpretation of the Cartesian operator space is illustrated in
Fig. 2.3. The identity operatorEis independent of rotation.
Single-transition shift operators can be defined in terms of the
Cartesian components or as products of kets and bras,
IþðÞ¼rs IxðÞþrs iIyðÞ¼rs jirhjs,
IðÞ¼rs IxðÞrs iIyðÞ¼rs jishjr:½ 2 : 212 As noted by Ernst ( 2 ), the indices are ordered such thatMr 4 Ms,as
defined by [2.139], to ensure that the raising and lowering operators
increase and decrease the magnetic quantum numbers, respectively:
IþðÞrsjis ¼jir, IðÞrsjir ¼jis: ½ 2 : 213 For the one-spin case, the eigenstates areji and
, and the density
operator can be expanded in terms of the basis operatorsI^ ,I^ ,Iþ,
andI. In this case, for example,Iji ¼
(^) h ji ¼
(^) because the
eigenstateji is associated with m¼þ1/2 and
(^) is associated with
m¼1/2. SimilarlyIþ
(^)
¼ji
(^)
¼ji.
The potential of the product operator approach becomes evident in
the case of two weakly scalar coupled spins. Each pair of spins has four
w
I
Ix
Iz
Iy
FIGURE2.3 Geometrical representations of rotations in an operator space:
precession of the angular momentum operator about the effective field direction
in angular momentum operator space.
2.7 PRODUCTOPERATORFORMALISM 81