QMGreensite_merged

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

ðpÞ¼expð
i IxÞð 0 Þexpði IxÞ: ½ 2 : 101 Š

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 Þ¼expð
i IxÞ, ½ 2 : 102 Š

then [2.101] becomes

ðtÞ¼Rxð Þð 0 ÞR
x^1 ð Þ: ½ 2 : 103 Š

The rotation operators can be expanded as

R
x^1 ð Þ¼Eþi Ix
^122 I^2 xþ...: ½ 2 : 104 Š

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 Š

I^2 n¼

1

4 n

E, ½ 2 : 106 Š

I^2 nþ^1 ¼

1

4 n

I: ½ 2 : 107 Š

Substituting the results contained in [2.105]–[2.107] into [2.104] and
grouping together even and odd powers ofiIxyields

R
x^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 Š

ExpandingIxin terms of the raising and lowering operators,

Iþ¼IxþiIy, I
¼Ix
iIy, ½ 2 : 109 Š

yields

2 Ix¼ðIþþI
ÞT: ½ 2 : 110 Š

52 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY