because [2.143] yields the results that
Iy¼
1
2
00 i 0
00 0i
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
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 JISIzSzÞt¼
eið^ Iþ^ SþJISÞt=^2 00 0
0 eið^ I^ SJISÞt=^200
00 eið^ Iþ^ SJISÞt=^20
00 0eið^ I^ SþJISÞt=^2
2
6
(^66)
4
3
7
(^77)
5
:
½ 2 : 186
Performing the matrix multiplications yields
exp½ið (^) IIzþ (^) SSzþ 2 JISIzSzÞtIySy
exp½ið (^) IIzþ (^) SSzþ 2 JISIzSzÞt
¼
i
2
0 eið^ SþJISÞteið^ IþJISÞt 0
eið^ SþJISÞt 00 eið^ IJISÞt
eið^ IþJISÞt 00 eið^ SJISÞt
0 eið^ IJISÞteið^ SJISÞt 0
2
(^66)
(^64)
3
(^77)
75 :
½ 2 : 187
This result is the final density operator(t). The observable signal is
found by forming the product with operatorFþ/IþþSþ,
i
2
0 eið^ SþJISÞt eið^ IþJISÞt 0
eið^ SþJISÞt 00 eið^ IJISÞt
eið^ IþJISÞt 00 eið^ SJISÞt
0 eið^ IJISÞt eið^ SJISÞt 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 0eið^ IþJISÞteið^ SþJISÞt
0 eið^ SþJISÞt eið^ SþJISÞt 0
0 eið^ IþJISÞt eið^ IþJISÞt 0
00 0 eið^ IJISÞtþeið^ SJISÞt
2
(^66)
(^64)
3
(^77)
75 :
½ 2 : 188
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 69