The trace of this result with the observation operator, ignoring for
convenience the constants of proportionality, yields
Trfexp½iHtIxexp½iHtFþg¼^12 exp½ið (^) IþJÞtþ^12 exp½ið (^) IJÞt:
½ 2 : 247
Both terms comprising the detected signal arepositive, indicating an
in-phasecomponent of thex-magnetization. The frequencies of the two
components of the in-phase signal are separated by the scalar coupling
between the two spins.An operator with a single transverse Cartesian
component is observable. Another example of a single-quantum
coherence operator is
2 IxSz¼ 21 ðIþS^ þIS^ Þ^12 ðIþS^ þIS^ Þ: ½ 2 : 248
Evolution under the free-precession Hamiltonian yields
expfiHtg 2 IxSzexpfiHtg¼ 21 ðIþS^ exp½ið (^) IþJÞt
þIS^ exp½ið (^) IþJÞtÞ
^12 ðIþS^ exp½ið (^) IJÞt
þIS^ exp½ið (^) IJÞtÞ: ½ 2 : 249
The trace of this result with the observation operator yields
Trfexp½iHt 2 IxSzexp½iHtFþg
¼^12 exp½ið (^) IþJÞt^12 exp½ið (^) IJÞt: ½ 2 : 250
In this case, the contributions from the Sspin are of opposite sign,
indicating anantiphase x-component of the magnetization on theIspin,
in which the two components of the signal have opposite sign. Formally,
the antiphase terms are not directly observable in the sense that, at a
particular instant, a term such as 2IxSz does not contribute to the
observed x-magnetization. However, antiphase operators will evolve
under the influence of the scalar coupling interaction, provided that
I andS have a nonzero scalar coupling constant, into an in-phase
operator that is detectable. Analogous terms for the Ispin product
operator involvingy-components are similar except that the phase of the
magnetization is shifted by 90 8.
The preceding detailed calculations are not necessary in practice,
because the form of the observable signal can be determined by
inspection of the coherences present at the start of the acquisition period.
For a system of N spins, the operators Iix and Iiy (1iN) are
2.7 PRODUCTOPERATORFORMALISM 89