QMGreensite_merged

Iy )

2 JISIzSzt

Iycos 2ðÞ
JISt 2 IxSzsin 2ðÞJISt, ½ 2 : 227 Š

Iz )

2 JISIzSzt

Iz: ½ 2 : 228 Š

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 Š

2 IySz )

2 JISIzSzt

2 IySzcos 2ðÞ
JISt Ixsin 2ðÞJISt, ½ 2 : 230 Š

2 IzSz )

2 JISIzSzt

2 IzSz: ½ 2 : 231 Š

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