QMGreensite_merged

Using the matrix representation given in [2.152], the following relation-
ship is easily derived

ð 2 IzSzÞ^2 n¼E: ½ 2 : 168 Š

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 Š

Again using [2.152], the matrix representation of the operator becomes

exp½i 2 IzSzŠ¼

cþis 000

0 c
is 00

00 c
is 0

000 cþis

2

(^66)
4
3
(^77)
5 , ½^2 :^170 Š
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
R
x^1 ðÞ¼
N
j¼ 1
R
jx^1 ðÞ¼
N
j¼ 1
Ecos
2
þisin
2
Tj

, ½ 2 : 171 Š
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,
R
x^1 ðÞ¼ Ecos
2
þisin
2
T 1

Ecos
2
þisin
2
T 2

: ½ 2 : 172 Š
The elements of the matrix representation ofRare constructed from the
basis eigenfunctions using the expressions
R
x^1 ðÞ


rs¼hjr^
N
j¼ 1
Ecos
2
þisin
2
Tj

jis: ½ 2 : 173 Š
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 65