Iy )
2 JISIzSzt
Iycos 2ðÞJISt 2 IxSzsin 2ðÞJISt, ½ 2 : 227
Iz )
2 JISIzSzt
Iz: ½ 2 : 228
Equations [2.226] and [2.227] demonstrate that single-spin operators
evolve into two-spin operators under the influence of the scalar coupling
interaction. The analogous evolution of the two-spin operators, 2ISz,
is given by
2 IxSz )
2 JISIzSzt
2 IxSzcos 2ðÞþJISt Iysin 2ðÞJISt, ½ 2 : 229
2 IySz )
2 JISIzSzt
2 IySzcos 2ðÞJISt Ixsin 2ðÞJISt, ½ 2 : 230
2 IzSz )
2 JISIzSzt
2 IzSz: ½ 2 : 231
Evolution of theSand 2IzSoperators is obtained by exchanging the
IandSlabels in [2.223]–[2.231].
2.7.3.2 Pulses On-resonance rf pulses applied along a specific axis
induce rotations in a plane orthogonal to that axis. The Hamiltonian
A
C
B
FIGURE2.5 Operator rotations. The rotations induced by an operator C
acting on an operatorAare illustrated. The operators satisfy the commutation
relationship [2.221] and the rotations are represented mathematically by [2.222].
2.7 PRODUCTOPERATORFORMALISM 85