QMGreensite_merged

in which the last line is obtained by using the Pauli spin matrices, [2.71].
If ¼ 1808 , the final matrix would be equal to

RxðÞIzR
x^1 ðÞ¼

1

2

 10

01



¼


1

2

10

0 
 1



¼
Iz, ½ 2 : 130 Š

corresponding to population inversion. If ¼ 908 , then the final matrix
becomes

Rxð= 2 ÞIzR
x^1 ð= 2 Þ¼

1

2

0 i


i 0



¼


1

2

0 
i

i 0



¼
Iy: ½ 2 : 131 Š

Simply, a 90 8 pulse applied withx-phase toIzmagnetization generates

Iymagnetization. These results are identical to the results obtained
using the Bloch model. The
Iy magnetization will evolve during
acquisition under the Zeeman Hamiltonian (in the rotating frame) given
as

Hz¼ðÞ! 0 
!rfIz¼ Iz: ½ 2 : 132 Š

This is a time-independent Hamiltonian; therefore,

ðÞ¼t expðÞ
iHztðÞ 0 expðÞiHzt

¼expðÞ
i IztðÞ 0 expðÞi Izt

¼UðÞ 0 U
^1 ½ 2 : 133 Š

and

U¼expðÞ¼
i tIz expð
 0 i^ t=^2 Þ expði^0 t= 2 Þ



: ½ 2 : 134 Š

Performing the matrix manipulations for(0)¼
Iyyields

ðtÞ¼

1

2

0 iexpð
i tÞ


iexpði tÞ 0



¼

1

2

0 i½Šcosð tÞ
isinð tÞ


i½Šcosð tÞþisinð tÞ 0



¼

1

2

0 icosð tÞþsinð tÞ


icosð tÞþsinð tÞ 0



¼

1

2

0 sinð tÞ

sinð tÞ 0



þ

i

2

0 cosð tÞ


cosð tÞ 0



: ½ 2 : 135 Š

2.4 QUANTUMMECHANICALNMR SPECTROSCOPY 57