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eigenstates increases unless implemented numerically on a computer.
Unfortunately as well, the density matrix formalism provides little
physical insight into NMR experiments. The design of new experiments
and the optimization of existing experiments are facilitated if the
spectroscopist has an intuitive feel for the evolution of the important
components of the density operator at each point in the experiment.
The aim of the theoretical analysis of NMR spectroscopy is
prediction of the outcome of experiments. The Hamiltonian is an
operator, and as has been stated previously, physically observable
quantities such as energy, position, and angular momentum are
represented in quantum mechanics by operators. Therefore, concentra-
tion on the operators themselves, rather than on the solutions to the
Schro ̈dinger equation, proves to be a powerful approach. As an
illustration, the analysis of the one-pulse experiment in Section 2.4.2
indicates that the equilibrium density operator can be expressed in terms
of the CartesianIzspin operator. This operator is partially converted
into theIy operator by a pulse withx-phase and rotation angle ;
subsequent evolution under the Zeeman Hamiltonian converts theIy
operator into a linear combination ofIxandIyspin operators. In this
case, the evolution of the density operator is represented by the
interconversion of single spin operators. Increasingly, due to the
continued development of stronger magnets, spin systems of interest
in heteronuclear and^1 H NMR spectroscopy of proteins are weakly
coupled. A simplified formalism, referred to as theproduct operator
formalism, that treats each weakly coupled system independently can be
used to analyze evolution of the density operator (9–11). The product
operator formalism retains much of the rigor of the full density matrix
treatment while facilitating manual computation and offering consider-
able insight into complex NMR experiments.

2.7.1 OPERATORSPACES
In general, an arbitrary density operator can be represented as a
linear combination of a complete set of orthogonal basis operators,Bk:

ðÞ¼t

XK

k¼ 1

bkðÞtBk, ½ 2 : 201 Š

in whichbk(t) are complex coefficients andKis the dimensionality of the
Liouville operator space spanned by the basis operators. For a system of
Nspin-1/2 nuclei,K¼ 4 N.Liouville operator space, and its attendant
operator algebra, can be regarded as an elaboration of the ideas of the

78 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY