QMGreensite_merged

ofR
x^1 ðÞ ,

RxðÞ¼

c^2 
icsu 
icsv 
s^2


icsu 1 
s^2 u^2 
s^2 uv 
icsu


icsv 
s^2 uv 1 
s^2 v^2 
icsv


s^2 
icsu 
icsv c^2

2

6

(^64)
3
7
(^75) : ½ 2 : 177 Š
The same calculation can be performed using rotation matrices that
concentrate on each spin in the two-spin system individually rather than
both at the same time. This approach can be particularly useful in
heteronuclear NMR experiments. The matrix representations of the
rotation operators are obtained from the direct products of the single-
spin rotation operators derived previously in [2.115–2.117]. For example,
for spinI,
RxðÞ ½Š¼I RxðÞ
E¼
cis c
is

(^1001)

¼
c 0
is 0
0 c 0
is

is 0 c 0
0
is 0 c
2
(^66)
4
3
(^77)
5 ,
½ 2 : 178 Š
RyðÞ ½Š¼I RyðÞ
E¼ csc
s

(^1001)

¼
c 0
s 0
0 c 0
s
s 0 c 0
0 s 0 c
2
6
(^64)
3
7
(^75) ,
½ 2 : 179 Š
and for spinS,
RxðÞ ½Š¼S E RxðÞ¼
10
01

c
is

is c

¼
c
is 00

is c 00
00 c
is
00
is c
2
(^66)
4
3
(^77)
5 ,
½ 2 : 180 Š
RyðÞ ½Š¼S E RyðÞ¼ 10
01

c
s
sc

¼
c
s 00
sc 00
00 c
s
00 sc
2
(^66)
4
3
(^77)
5 :
½ 2 : 181 Š
The resultRx( )¼Rx( )[I]Rx( )[S] is obtained by matrix multiplication
and agrees with [2.176] in the weak coupling limit where¼0.
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 67