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
RxðÞIzRx^1 ðÞ¼
1
2
10
01
¼
1
2
10
0 1
¼Iz, ½ 2 : 130
corresponding to population inversion. If ¼ 908 , then the final matrix
becomes
Rxð= 2 ÞIzRx^1 ð= 2 Þ¼
1
2
0 i
i 0
¼
1
2
0 i
i 0
¼Iy: ½ 2 : 131
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
This is a time-independent Hamiltonian; therefore,
ðÞ¼t expðÞiHztðÞ 0 expðÞiHzt
¼expðÞi IztðÞ 0 expðÞi Izt
¼UðÞ 0 U^1 ½ 2 : 133
and
U¼expðÞ¼i tIz expð 0 i^ t=^2 Þ expði^0 t= 2 Þ
: ½ 2 : 134
Performing the matrix manipulations for(0)¼Iyyields
ðtÞ¼
1
2
0 iexpði tÞ
iexpði tÞ 0
¼
1
2
0 i½cosð tÞisinð tÞ
i½cosð 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
2.4 QUANTUMMECHANICALNMR SPECTROSCOPY 57