Section 2.3. These solutions can be applied as a recipe by using a simple
set of rules, which are presented in Section 2.7.3. The Hamiltonians
of most interest in solution NMR consist of one or more of four
interactions: (1) rf pulse, (2) chemical shift, (3) scalar coupling, and
(4) residual dipolar coupling. Most importantly, in the weak coupling
regime, the chemical shift, scalar, and residual dipolar coupling inter-
actions commute with each other. Note that throughout this analysis
relaxation of the spins back to equilibrium is not considered.
2.7.2 BASISOPERATORS
The choice of basis operators is determined by the problem at hand
at any specific time. For example, the angular momentum operators
Ix,Iy, andIz, which represent thex-,y-, andz-components of the spin
angular momentum of the system, are particularly useful for calculating
the effects of rf pulses, whereas the shift operators,Iþ andI, are
particularly suited to evaluating evolution under the free-precession
Hamiltonian. For asingle spin, the state of a magnetization vector can be
specified by the amounts ofx,y, andzmagnetization. In the same way,
the quantum mechanical state of the system can be described by
specifying the magnitudes of the operators that are present at any time.
Formally, the state of the system is specified by the density operator and
the density operator is expressed as a linear combination of operators.
In most cases, Cartesian basis operators (E/2,Ix,Iy,Iz), will be employed.
Other basis sets, such as the single-element (I^ ,I^ ,Iþ,I) basis
operators, defined using the Dirac notation as
I^ ¼ji hj , Iþ¼ji(^)
,
I^ ¼
(^)
, I¼
hj ,
½ 2 : 209
and the shift basis operators ([2.191]), are also useful. The Cartesian and
single-element basis sets are related by
Iz¼^12 I^ I^
, Ix¼^12 ðÞIþþI,
Iy¼ 21 iðÞIþI,^12 E¼^12 I^ þI^
:
½ 2 : 210
Levitt notes that the three Cartesian operators form a three-
dimensional space in which the density operator, represented by a
vectorI, rotates at a frequencyjxj¼ðxxÞ^1 =^2 about a vectorx, with
Cartesian components
!x¼! 1 cos , !y¼! 1 sin , !z¼ , ½ 2 : 211
80 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY