Using the matrix representation given in [2.152], the following relation-
ship is easily derived
ð 2 IzSzÞ^2 n¼E: ½ 2 : 168
Substituting the results contained in [2.168] into [2.167] and grouping
together even and odd powers ofiIzSzyields
expði 2 IzSzÞ¼E 1
2
2! 22
þ
4
4! 24
þ
þ 4 iIzSz
2
3
3! 23
þ
5
5! 25
þ
¼Ecos
2
þ 4 iIzSzsin
2
:
½ 2 : 169
Again using [2.152], the matrix representation of the operator becomes
exp½i 2 IzSz¼
cþis 000
0 cis 00
00 cis 0
000 cþis
2
(^66)
4
3
(^77)
5 , ½^2 :^170
wherec¼cos( /2) ands¼sin( /2).
2.5.3 ROTATIONS INPRODUCTSPACES
For a homonuclear system ofNspins, the matrix representation of
the pulse operator can be calculated from
Rx^1 ðÞ¼
N
j¼ 1
Rjx^1 ðÞ¼
N
j¼ 1
Ecos
2
þisin
2
Tj
, ½ 2 : 171
in which ¼ B 1 p. In [2.171], the effect of the scalar coupling term of
the Hamiltonian has been ignored; this simplification requires that the
length of the rf pulse,p, satisfy 2Jijp1. For a two-spin system,
Rx^1 ðÞ¼ Ecos
2
þisin
2
T 1
Ecos
2
þisin
2
T 2
: ½ 2 : 172
The elements of the matrix representation ofRare constructed from the
basis eigenfunctions using the expressions
Rx^1 ðÞ
rs¼hjr^
N
j¼ 1
Ecos
2
þisin
2
Tj
jis: ½ 2 : 173
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 65