matrix representation of the operator,H^0 , given by
H^0 ¼UHU^1 , ½ 2 : 92 is a diagonal matrix. The transformed basis functions given by [2.78]
then represent the eigenfunctions of the Hamiltonian operator and the
diagonal elements ofH^0 are the energies associated with the stationary
states of the system.
2.3 Pulses and Rotation Operators
The simple case of applying an rf pulse to a single spin-1/2 nucleus
in a static fieldB 0 will be considered first. The pulse is applied as
a linearly polarized transverse rf field with magnitude 2B 1 and angular
frequency!rf. Remembering from the Bloch approach that this field
can be decomposed into two counter-rotating fields, only one of which
has a significant effect on the spin, the Hamiltonian for the pulse is
written as ( 8 )
H¼lBðtÞ¼HzþHrf
¼! 0 Izþ! 1 ½Ixcosð!rftþ ÞþIysinð!rftþ Þ, ½ 2 : 93 where I is the spin angular momentum operator along the axis ,
! 0 ¼ B 0 , and! 1 ¼ B 1. The first term in [2.93] takes into account the
precession of the spin under the influence of the static field, that is, the
Zeeman Hamiltonian, and the second term represents the pulse.
The choice ofUthat removes the time dependence from [2.93] is
U¼expði!rfIztÞ: ½ 2 : 94
Application of this unitary transformation, using [2.66], gives the
effective Hamiltonian,
He¼! 0 Izþ! 1 expði!rfIztÞ½Ixcosð!rftþ ÞþIysinð!rftþ Þexpði!rfIztÞ
þiexpði!rfIztÞi!rfIzexpði!rfIztÞ:
½ 2 : 95
Using the rotation properties of the spin angular momentum operators
presented in Table 2.1 (these properties will be derived later),
Ixcosð!rftÞþIysinð!rftÞ¼expði!rfIztÞIxexpði!rfIztÞ
Ixsinð!rftÞþIycosð!rftÞ
¼expði!rfIztÞIyexpði!rfIztÞ, ½ 2 : 96 50 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY