2.7.3 EVOLUTION IN THEPRODUCTOPERATORFORMALISM
The goal of the product operator formalism is to derive the evolution
of a spin system through a particular pulse sequence as conveniently
as possible. Effects of pulses and delays in terms of Cartesian product
operators are extremely simple, because each factor of the product is
rotated independently. Rotation operator equations similar to [2.136]
can be derived by the matrix derivations established previously; however,
this approach is rather laborious. Instead, the rules for transformations
of product operators can be established using the following useful
theorem: if three operators satisfy the commutation relationship (and its
cyclic permutations)
½¼A,B iC, ½ 2 : 221 then
expðÞiCAexpðÞ¼iC AcosþBsin: ½ 2 : 222 Equation [2.222] can be verified by differentiating exp(–iC)Aexp(iC)
twice with respect to, applying the commutation relations and solving
the resulting harmonic differential equation. The evolution indicated by
[2.222] can be illustrated succinctly by Fig. 2.5.
2.7.3.1 Free Precession During periods of free precession, the
effects of chemical shift evolution and scalar coupling evolution must be
considered. For a spinI, the chemical shift Hamiltonian has the form
H¼ (^) IIz, where (^) Iis the offset of spinI. Evolution during a delay,t,
is described by
Ix )
(^) IIzt
IxcosðÞþ (^) It IysinðÞ (^) It, ½ 2 : 223
Iy^ IIzt )IycosðÞ (^) It IxsinðÞ (^) It, ½ 2 : 224
Iz )Iz:
(^) IIzt
½ 2 : 225
For a weakly coupled two-spin system,IandS, the scalar coupling
Hamiltonian has the form H¼ 2 JISIzSz, where JIS is the scalar
coupling constant. Evolution of the single-spin operators during a delay,
t, is described by
Ix )
2 JISIzSzt
Ixcos 2ðÞþJISt 2 IySzsin 2ðÞJISt, ½ 2 : 226
84 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY