QMGreensite_merged

representation ofIzþSzin a two-spin system must be a 44 matrix
because the vector space is spanned by four wavefunctions; thus,

IzþSz 6 ¼

1

2

10

0 
 1



þ

1

2

10

0 
 1



¼^100 
 1



: ½ 2 : 142 Š

A more formal analysis indicates that matrix representations of the
operators in the two-spin system can be calculated from the direct
product of the one-spin operators with the identity operator. The results
for a two-spin system are

Ið2spinÞ¼Ið1spinÞ E and Sð2spinÞ¼E Sð1spinÞ; ½ 2 : 143 Š

where¼x,y,orz. In general, for an N-spin system, the representations
of the angular momentum operators for thekth spin are given by

IðkNspinÞ¼E 1

 E 2

 Ek
 1

 Iðk1spin Þ Ekþ 1

 EN: ½ 2 : 144 Š

Returning to the previous example,

Izð2spinÞ¼Izð1spinÞ E¼

1

2

10

0 
 1



10

01



¼

1

2

10 0 0

01 0 0

00 
 10

00 0
 1

2

(^66)
4
3
(^77)
5 ,
½ 2 : 145 Š
Szð2spinÞ¼E Szð1spinÞ¼
10
01

1
2
10
0
1

¼
1
2
1000
0
10 0
0010
000
1
2
(^66)
4
3
(^77)
5 :
½ 2 : 146 Š
The combination of Izð2spinÞþSzð2spinÞ gives the correct matrix
representation:
Izð2spinÞþSzð2spinÞ¼
100 0
000 0
000 0
000
1
2
6
(^64)
3
7
(^75) : ½ 2 : 147 Š
From now on, the (2spin) superscript will be implied. The fundamental
rule of the operator algebra in direct product spaces is (as illustrated for
60 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY