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
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
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
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
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
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
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