QMGreensite_merged

the second term in [2.95] is simplified to! 1 (IxcosþIysin). The third
term in [2.95] is simplified to
!rfIz because an operator commutes
with an exponential operator of itself. The effective Hamiltonian can be
written as

He¼! 0 Izþ! 1 ðIxcosþIysinÞ
!rfIz

¼ð! 0 
!rfÞIzþ! 1 ðIxcosþIysinÞ

¼ Izþ! 1 ðIxcosþIysinÞ: ½ 2 : 97 Š

This is now atime-independenteffective Hamiltonian and the solution
in the form of [2.67] describes evolution of the density operator in the
rotating frame. Note the strong similarity between [2.97] and [1.18].
For completeness, the isotropic chemical shift Hamiltonian is given by

H¼
! 0 Iz, ½ 2 : 98 Š

in whichis the isotropic shielding constant [1.48], rather than the
density operator, and can be incorporated into the definition of
¼! 0 (1
)
!rf.
If ¼0 and ¼0, then the Hamiltonian for an on-resonance
x-pulse becomes

He¼! 1 Ix ½ 2 : 99 Š

and, as follows from [2.67],

ðpÞ¼expð
iHepÞð 0 ÞexpðiHepÞ

¼expð
i! 1 IxpÞð 0 Þexpði! 1 IxpÞ: ½ 2 : 100 Š

For simplicity, the superscript has been omitted from the rotating frame
density operator; in general, context is sufficient to establish whether a

TABLE2.1

Rotation Properties of Angular Momentum Operators

u,va xyz

xIx Ixcos–Izsin IxcosþIysin
yIycosþIzsin Iy Iycos–Ixsin
zIzcos–Iysin IzcosþIxsin Iz

aThe table entries (u,v) are the results of the unitary transformation exp(–iIv)Iuexp(iIv).

2.3 PULSES ANDROTATIONOPERATORS 51