h jj i¼h jj i¼0, and there isno coherence between the two states.
Therefore, as has been stated previously, the equilibrium density matrix
is diagonal.
The application of an rf pulse to the equilibrium density operator
induces exchange of population (i.e., transitions) between stationary
states for whichm¼ 1 and causes perturbations of the equilibrium
population distribution. In the case of a spin-1/2 nucleus, an rf pulse that
redistributes populations across the $ transition creates a phase
relationship across that transition such that expið Þ
6 ¼ 0
(averaged over the ensemble and assuming that the rotation angle
is not a multiple of 180 8 ). The density operator following the pulse is
said to represent a coherent superposition between the two states;
more commonly, this phenomenon is referred to simply ascoherence.
Coherence describes correlation of quantum mechanical phase relation-
ships among a number of systems (separate nuclei) that persists even
after the rf field is removed. Coherence is a phenomenonassociatedwith
an NMR transition and is not a transition itself; evolution of coherence
doesnotchange the populations of the spin states. Nonzero off-diagonal
elements of the density matrix denote the existence of coherence.
Both shift and Cartesian basis operators are useful for describing
NMR spectroscopy. The Cartesian operators are a convenient basis for
describing the effects of rf pulses on the density operator, and the shift
operators are a convenient basis for describing the evolution of
coherence in an NMR experiment. Only two eigenstates,j iandj i,
exist for a single spin-1/2 nucleus; consequently, coherences associated
with thej i$j itransitions with m¼ 1 are conveniently repre-
sented by the raising and lowering operatorsIþandI. Four eigenstates
exist for a two-spin system. Figure 2.1 illustrates the appearance of
double- and zero-quantum coherence where eigenstates are connected in
whichm¼ 2 andm¼0, respectively. Double-quantum coherence is
associated with transitions in which the spin states can change from
$. The change in eigenstate is identical for both of the spins
involved, and this is often called a ‘‘flip-flip’’ transition. On the other
hand, zero-quantum coherence is associated with transitions in which the
spin states change $ , i.e., in the opposite sense to each other;
these are often called ‘‘flip-flop’’ terms.
The two-spin case will be seen to be the most commonly encountered
as far as this text is concerned; however, spin systems consisting of
three or more scalar coupled spins are evidently important and display
additional features. Some of the salient features of larger spin systems
will be briefly discussed using a weakly coupled three-spin system as an
exemplar. In the two-spin case, each of them¼ 1 transitions involves
2.6 COHERENCE 75