The preceding examples illustrate the concept ofcoherence, which is
one of the most fundamental aspects of NMR spectroscopy. As has been
stated previously, a diagonal matrix element of the density operator,
nn¼cncn, is a real, positive number that corresponds to the population
of the state described by the basis functionjin. Formally, an off-diagonal
element of the density operator,nm, represents coherence between
eigenstatesjin andjim, in the sense that the time-dependent phase
properties of the various members of the ensemble are correlated with
respect tojin andjim. Those matrix elements that denotem¼ 1 are
calledsingle-quantum coherence; those denoting m¼ 2 are called
double-quantum coherence and, not surprisingly, those denoting
m¼0 are calledzero-quantum coherence.
To make these ideas more concrete, consider the following example.
The coefficientscnfor the two-level system for a spin-1/2 can be written
in polar notation in terms of an amplitude and a phase factor for the
and states,
c ¼jjc expðÞi , ½ 2 : 196 c ¼ c(^)
expi
: ½ 2 : 197
Any wavefunction can be expressed as
¼c ji þc
(^)
¼jjc expðÞi ji þ c
(^)
expi
(^)
, ½ 2 : 198
thus, for a pure state, the matrix elements of the projection operator
P¼jihj are
hj Pji ¼jjc 2 , hj P
¼jjc c
(^)
expi
,
(^)
P
(^) ¼ c
(^2) ,^
(^) Pji ¼jjc c
(^) exp i :
½ 2 : 199
Because the state is pure, all members of the ensemble are identical and
the terms ( – ) do not vary between members of the ensemble. For a
mixed, macroscopic state, however,
hj ji¼jjc 2 , hj
¼jjc c
(^)
expið Þ
,
(^)
¼ c
(^2)
,
(^)
ji¼jjc c
(^)
expið Þ
:
½ 2 : 200
If no relationship exists between the macroscopic phase properties of
the state (across the ensemble) and the phase properties of the state
(across the ensemble), then ( ) takes on all values in the range
0to2, and exp½ið Þ¼exp½ið Þ¼0. In this case,
74 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY