The trace of this matrix is proportional to the observed complex
magnetization:
Mþ
ðtÞ/eið^ IþJISÞtþeið^ IJISÞtþeið^ SþJISÞtþeið^ SJISÞt: ½ 2 : 189 The spectrum consists of four signals arranged into two doublets.
One doublet consists of the frequencies (^) I JISand the other doublet
consists of the frequencies (^) S JIS.
2.6 Coherence
So far the density operator has been represented in terms of a
Cartesian basis of the spin angular momentum operatorsIx,Iy, andIz.
Product operators in the Cartesian basis will be used most often in this
text because the Cartesian basis affords the simplest treatment of pulses
during a pulse sequence (9–11). For a system of two spin-1/2 nuclei,
16 Cartesian product operator terms are required:
ðÞ 1 = 2 E Ix Iy Iz Sx Sy Sz
2 IxSz 2 IySz 2 IzSz 2 IzSx 2 IzSy
2 IxSx 2 IySy 2 IxSy 2 IySx
½ 2 : 190
The matrix representations of these two-spin product operators, derived
using [2.149], are shown in Table 2.2.
The density operator also can be expressed in the shift operator
basis, which provides additional insight into the density matrix theory.
For a single spin-1/2 nucleus, the shift basis consists of the operators
Iþ¼IxþiIy¼
01
00
, I¼IxiIy¼
00
10
,
I 0 ¼
ffiffiffi
2
p
Iz¼
1
ffiffiffi
2
p
10
0 1
,
1
ffiffiffi
2
p E¼
1
ffiffiffi
2
p
10
01
,
½ 2 : 191
formed by taking linear combinations of the Cartesian operators. As
discussed in Section 2.7.1, the factors of 2–1/2appearing in the matrix
representations of the operators are normalization factors. Operators in
the shift basis are transformed to the Cartesian basis by
Ix¼^12 IþþI
, Iy¼ 21 iIþI
, Iz¼p^1 ffiffi 2 I 0 : ½ 2 : 192
70 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY