QMGreensite_merged

In isotropic solution, the Zeeman, chemical shift, scalar coupling, and rf
pulse terms are the dominant interactions. The expectation value of the
observed signal at the desired time is calculated using [2.47] as the trace
of the product of the density operator and the observation operator
corresponding to the observable magnetization. The equilibrium density
and observation operators are described in the following section.

2.4.1 EQUILIBRIUM ANDOBSERVATIONOPERATORS
The lattice is assumed to always be in thermal equilibrium at a
temperatureT(equivalently, the lattice is assumed to have infinite heat
capacity). At thermodynamic equilibrium, the nuclear spin states are
assumed to be in thermal equilibrium with the lattice. Consequently,
the values ofP(
) (see Section 2.2.2) are constrained such that the
populations of the stationary states (given by the diagonal elements of
the density matrix) have a Boltzmann distribution. Furthermore, the
density matrix is diagonal at equilibrium because the members of the
different subensembles described byP(
) are uncorrelated. The form of
the equilibrium density operator that satisfies these requirements is

eq¼e
H=kBT=Tr e
H=kBT



: ½ 2 : 123 Š

In the eigenbase of the Hamiltonian operator, the matrix elements ofeq
are given by

mneq¼hjme
H=kBTjin

XN

i¼ 1

hjie
H=kBTjii

¼m,ne
En=kBT

XN

i¼ 1

e
Ei=kBT, ½ 2 : 124 Š

which is a diagonal matrix whose elements are the required Boltzmann
probabilities. In the high-temperature approximation,EnkBTand the
equilibrium density operator can be approximated by

eq¼e
H=kBT=Tre
H=kBT



 E
H=kBT



=TrE
H=kBT



 E
H=kBT



=TrfgE

E=N
H=ðNkBTÞ, ½ 2 : 125 Š

in whichNis the dimensionality of the Hilbert space and is equal to 2M
forMspin-1/2 nuclei (i.e.,M¼2 for anIStwo-spin system). The term

2.4 QUANTUMMECHANICALNMR SPECTROSCOPY 55