For example, using the strong coupling eigenbasis [2.158], the matrix
element½Rx^1 ðÞ 12 is calculated as
Rx^1 ðÞ
12 ¼hj^1
N
j¼ 1
Ecos^
2
þisin^
2
Tj
ji 2
¼hj
N
j¼ 1 Ecos
2 þisin
2 Tj
cos
þsin
¼hj Ecos^2
2
þicos^
2
sin^
2
T 1 þicos^
2
sin^
2
T 2 sin^2
2
T 1 T 2
cos
þsin
¼icos^
2
sin^
2
sinþicos^
2
sin^
2
cos
¼icos^2 sin^2 ðÞcosþsin:
½ 2 : 174
This result is calculated using the property that the inversion operatorTj
changes the spin state of spinjfrom to andvice versa. As another
example, Rx^1 ðÞ
14 is given by
Rx^1 ðÞ
14 ¼hj^1
N
j¼ 1
Ecos^
2
þisin^
2
Tj
ji 4
¼hj
N
j¼ 1
Ecos
2 þisin
2 Tj
¼hj Ecos^2
2 þicos
2 sin
2 T^1 þicos
2 sin
2 T^2 sin
2
2 T^1 T^2
¼sin^2
2 :
½ 2 : 175
Repeating these calculations for every element of the matrix representa-
tion of the pulse operator yields
Rx^1 ðÞ¼
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 : 176
where c¼cos( /2), s¼sin( /2), u¼cosþsin, and v¼cos– sin.
Because the rotation operators are unitary, Rx( ) is the adjoint
66 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY