QMGreensite_merged

The 16 operators in the shift basis for a system of two spin-1/2 nuclei are
ðÞ 1 = 2 E Iþ I
I 0 Sþ S
S 0
IþS 0 I
S 0 I 0 S 0 I 0 Sþ I 0 S

IþSþ 2 I
S
2 IþS
2 I
Sþ

½ 2 : 193 Š

The matrix representations of these operators are shown in
Table 2.3. These operators are constructed from the direct products of
the respective operators for each individual spin. For example,

I
¼

1

ffiffiffi

2

p

0000

0000

1000

0100

2

(^66)
4
3
(^77)
5 , I
þSþ¼
0001
0000
0000
0000
2
(^66)
4
3
(^77)
5 ,
IþS
¼
0000
0010
0000
0000
2
(^66)
4
3
(^77)
5 :
½ 2 : 194 Š
The physical meaning of the shift operator basis is illustrated by
examining the matrix representations in Table 2.3. First, consider theI

operator. For illustration, the matrix is written in [2.195] with the spin
states of the system along the side and top of the matrix to indicate the
spin states connected by each matrix element,
I
¼
(^1) ffiffiffi
2
p
0000
0000
1000
0100
2
(^66)
(^64)
3
(^77)
(^75)
(^)
(^)
(^)
(^)
: ½ 2 : 195 Š
The only nonzero matrix elements present correspond to the transitions
! and!. The lowering operator,I
, is associated with
a change in the spin angular momentum quantum number ofm¼–1
and a change in the state of theIspin from (þ1/2)! (
1/2). In the
case of theIþSþoperator, the only nonzero matrix element corresponds
to m¼þ2 and a change in spin state from
(^) !ji. In this
instance, both theIspin and theS spin change state from to.
Similarly for theIþS
operator, the nonzero matrix element corre-
sponds to the transition
(^)
!
(^)
withm¼0. In this example, both
spins change spin states in opposite senses.
72 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY