QMGreensite_merged

transitions 1–2 and 2–4). The values ofmifor the stationary states are
m 1 ¼þ3/2,m 2 ¼þ1/2, andm 4 ¼
1/2. The states represented bym 1 and
m 4 are at opposite ends of the transition pathway under discussion and
differ in their value ofmby 2. In this case, the two connected transitions
are said to be progressively connected. On the other hand, levels 6
and 7 in the connected transitions 5–6 and 5–7 do not differ in their
values ofm(e.g.,m 6 ¼m 7 ¼
1/2). In this instance, the transitions are
said to beregressivelyconnected. In contrast to the single- and two-spin
systems, single-quantum transitions exist in three-spin systems for which
all three nuclei change spin state. For example, the transition connecting
eigenstate 2 with eigenfunction
, and eigenstate 7 with eigenfunc-
tion
, hasm¼
1/2
1/2¼
1.

2.7 Product Operator Formalism

Although the density matrix theory provides a rigorous description
of the evolution of a nuclear spin system, the requisite matrix
calculations quickly become cumbersome as the number of spins and

aaa

baa

aba

aab

bbb

bba

bab

abb

JRS JRS

JRS JIR JIR

wS wR wI

JRS JRS JRS JIR

b

c

a

JIR

1

2

4

8

3

5

7

(^6) 1-5
2-6
3-7
4-8
1-2
5-6
3-4
7-8
6-85-72-41-3
FIGURE2.2 Spin states and spectrum for a three-spinIRSsystem. (a) The eight
spin states and the allowed single-quantum transitions between states are shown.
(- - -) Single-quantum transitions of theIspin, (—) single-quantum transitions of
theRspin, and (– - –) single-quantum transitions of theSspin. (b) A schematic
spectrum for anIRSspin system is shown for the special case thatJIR 5 JRSand
JIS¼0. (c) A schematic spectrum for anIRSspin system is shown for the special
case thatJIR¼JRSandJIS¼0.
2.7 PRODUCTOPERATORFORMALISM 77