QMGreensite_merged

2.5.4 ONE-PULSEEXPERIMENT FOR ATWO-SPINSYSTEM
To compute the observable magnetization following a pulse and
subsequent free precession, the evolution of the density operator,
beginning with the equilibrium matrix representation of the density
operator for a two-spin system, must be determined. Using [2.125], the
initial density matrix is written as

ð 0 Þ!IIzþ!SSz¼

1

2

!Iþ!S 00 0

0 !I
!S 00

00 
!Iþ!S 0

00 0
!I
!S

2

6

(^64)
3
7
(^75) ,
½ 2 : 182 Š
in which a common divisor of 2kBThas not been written for convenience
and weak coupling has been assumed. A pulse (^) x(with rotation angle
andx-phase) rotates an initial state of the density operator according to
the now well-known general equation,
ðtÞ¼RxðÞ ð 0 ÞR
x^1 ðÞ : ½ 2 : 183 Š
For simplicity, an ideal 90 8 pulse withx-phase will be assumed. Using
[2.182], [2.183], [2.176], and [2.177],
ðtÞ¼RxðÞ= 2 ð 0 ÞR
x^1 ðÞ= 2
¼
1
8
1
i
i
1

i 1
1
i

i
11
i

1
i
i 1
2
(^66)
(^64)
3
(^77)
(^75)
!Iþ!S 00 0
0 !I
!S 00
00
!Iþ!S 0
00 0
!I
!S
2
(^66)
(^64)
3
(^77)
(^75)

1 ii
1
i 1
1 i
i
11 i

1 ii 1
2
(^66)
6
4
3
(^77)
7
5
¼
1
2
0 i!S i!I 0

i!S 00 i!I

i!I 00 i!S
0
i!I
i!S 0
2
(^66)
(^64)
3
(^77)
75 ¼
!IIy
!SSy
½ 2 : 184 Š
68 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY