QMGreensite_merged

a two-spin system)

ABij

ðÞA B jii j



¼Ajii Bj

, ½ 2 : 148 Š

in whichAis an operator that acts on theispin andBis an operator that
acts on thejspin. Also note that

ABA B¼ðA EÞðE BÞ: ½ 2 : 149 Š

Thus, for example,

Iz
ðÞIz E ji



¼Izji E

¼

1

2

ji

¼

1

2

: ½ 2 : 150 Š

As a second example,

2 IzSz

(^)  2 ðÞIz Sz ji
 (^) ¼ 2 Izji Sz

¼ 2
1
2
ji


1
2
 (^)
¼

1
2
(^)
: ½ 2 : 151 Š
In matrix notation,
2 IzSz 2 Iz Sz¼
1
2
10
0
1

(^100)
1

¼
1
2
10 00
0
100
00
10
00 01
2
(^66)
4
3
(^77)
5 ,
½ 2 : 152 Š
so that [2.151] also can be written as
2 IzSz
(^) ^1
2
10 00
0
100
00
10
00 01
2
(^66)
4
3
(^77)
5
0
1
0
0
2
(^66)
4
3
(^77)
5 ¼
1
2
0

1
0
0
2
(^66)
4
3
(^77)
5 ¼

1
2
(^) : ½ 2 : 153 Š
As will be discussed in Section 2.7.1, the factor of 2 in the operator 2IzSz
is introduced as a normalization factor.
2.5.2 SCALARCOUPLINGHAMILTONIAN
The free-precession laboratory frame Hamiltonian for N scalar
coupled spins is
H 0 ¼HzþHJ¼
XN
i¼ 1
xiIizþ 2 p
XN
^1
i¼ 1
XN
j¼iþ 1
JijIiIj, ½ 2 : 154 Š
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 61