QMGreensite_merged

because [2.143] yields the results that

Iy¼

1

2

00 
i 0

00 0
i

i 00 0

0 i 00

2

(^66)
4
3
(^77)
5 and Sy¼
1
2
0
i 00
i 000
000
i
00 i 0
2
(^66)
4
3
(^77)
5 : ½^2 :^185 Š
This is exactly the expected result: each term in the initial density
operator is transformed identically by the nonselective pulse. Following
the pulse, the density operator evolves under the free-precession
Hamiltonian. Combining [2.134] with [2.170] yields the matrix repre-
sentation of the exponential operator,
exp½ið (^) IIzþ (^) SSzþ 2 JISIzSzÞtŠ¼
eið^ Iþ^ SþJISÞt=^2 00 0
0 eið^ I
^ S
JISÞt=^200
00 eið
^ Iþ^ S
JISÞt=^20
00 0eið
^ I
^ SþJISÞt=^2
2
6
(^66)
4
3
7
(^77)
5
:
½ 2 : 186 Š
Performing the matrix multiplications yields
exp½
ið (^) IIzþ (^) SSzþ 2 JISIzSzÞtŠ
Iy
Sy

exp½ið (^) IIzþ (^) SSzþ 2 JISIzSzÞtŠ
¼
i
2
0 e
ið^ SþJISÞte
ið^ IþJISÞt 0

eið^ SþJISÞt 00 e
ið^ I
JISÞt

eið^ IþJISÞt 00 e
ið^ S
JISÞt
0
eið^ I
JISÞt
eið^ S
JISÞt 0
2
(^66)
(^64)
3
(^77)
75 :
½ 2 : 187 Š
This result is the final density operator(t). The observable signal is
found by forming the product with operatorFþ/IþþSþ,
i
2
0 e
ið^ SþJISÞt e
ið^ IþJISÞt 0

eið^ SþJISÞt 00 e
ið^ I
JISÞt

eið^ IþJISÞt 00 e
ið^ S
JISÞt
0
eið^ I
JISÞt
eið^ S
JISÞt 0
2
(^66)
(^64)
3
(^77)
(^75)
0
1
10
00 0
1
00 0
1
0000
2
(^66)
(^64)
3
(^77)
(^75)
¼
i
2
00 0
e
ið^ IþJISÞt
e
ið^ SþJISÞt
0 eið^ SþJISÞt eið^ SþJISÞt 0
0 eið^ IþJISÞt eið^ IþJISÞt 0
00 0 eið^ I
JISÞtþeið^ S
JISÞt
2
(^66)
(^64)
3
(^77)
75 :
½ 2 : 188 Š
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 69