rotating frame or laboratory frame density operator is intended. If
¼! 1 p, is defined to be the flip angle of the pulse of lengthp, then
ðpÞ¼expði IxÞð 0 Þexpði IxÞ: ½ 2 : 101
The matrix representation of the exponential operators in [2.101] must
be derived so that the effect on the density operator can be calculated.
If the exponential rotation operators are defined as
Rxð Þ¼expði IxÞ, ½ 2 : 102
then [2.101] becomes
ðtÞ¼Rxð Þð 0 ÞRx^1 ð Þ: ½ 2 : 103
The rotation operators can be expanded as
Rx^1 ð Þ¼Eþi Ix^122 I^2 xþ...: ½ 2 : 104
Using the Pauli spin matrices given in [2.71], the following relationships
are easily derived:
I^2 x¼I^2 y¼I^2 z¼
1
4
E, ½ 2 : 105
I^2 n¼
1
4 n
E, ½ 2 : 106
I^2 nþ^1 ¼
1
4 n
I: ½ 2 : 107
Substituting the results contained in [2.105]–[2.107] into [2.104] and
grouping together even and odd powers ofiIxyields
Rx^1 ð Þ¼E 1
2
2! 22
þ
4
4! 24
þ
þ 2 iIx
2
3
3! 23
þ
5
5! 25
þ
¼EcosðÞþ= 2 2 iIxsinðÞ= 2 :
½ 2 : 108
ExpandingIxin terms of the raising and lowering operators,
Iþ¼IxþiIy, I¼IxiIy, ½ 2 : 109
yields
2 Ix¼ðIþþIÞT: ½ 2 : 110
52 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY