QMGreensite_merged

The trace of a product of matrices is invariant to cyclic permutations
of the matrices. Thus,

Trfg¼ABC Trfg¼CAB TrfgBCA: ½ 2 : 43 Š

A corollary of this theorem is that the trace of a commutator is zero:

Tr½ŠA,B



¼Trfg¼AB
BA Trfg
AB Trfg¼BA 0 : ½ 2 : 44 Š

2.2.2 QUANTUMSTATISTICALMECHANICS
The preceding analysis is applicable to a system in a so-called pure
state in which the entire system is described by the same wavefunction.
The wavefunction for a macromolecule in an NMR solution is an
enormously complicated function of the degrees of freedom of the
molecule and includes contributions from the spin, rotational, vibra-
tional, electronic, and translational properties of the molecule.
Determining the complete wavefunction for the molecule is both
unfeasible and unnecessary because the properties of the nuclear spins
are of primary interest in NMR spectroscopy. Accordingly, the system is
divided into two components: the spin system and the surroundings (i.e.,
all other degrees of freedom). For historical reasons, the surroundings
are termed thelattice. As a result of this division, the spin wavefunctions
for different molecules in the NMR sample are no longer identical, but
rather depend upon the ‘‘hidden’’ lattice variables. Such a system is
called amixed stateand the effects of the lattice are incorporated by
using statistical mechanics (2, 7). Each subensemble comprising the
sample can be described by a wavefunction,
, and a probability density,
P(
), that represents the contribution of the subensemble to the mixed
state. The statistical value of the expectation value for a mixed state is
then obtained by averaging over the probability distribution,

hAi¼

Z

Pð
Þhj
Aji
d

¼

X

nm

Z

Pð
ÞcncmdhjmAjin

¼

X

nm

cncmhjmAjin: ½ 2 : 45 Š

The factors cncm will vary from system to system, but the matrix
elementshjmAjin will not. An overbar has been used to denote the
statistical ensemble average in [2.45].

40 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY