QMGreensite_merged

in whichnis a unit vector along the direction of the effective field in the
rotating frameBr, given by [1.18], is given by [1.23],

nI¼IxcossinþIysinsinþIzcos, ½ 2 : 119 Š

andis defined by [1.21]. Rather than derive a matrix representation of
[2.118], the following identity will be established:

RðÞ¼ , RzðÞRyðÞRzðÞ R
y^1 ðÞR
z^1 ðÞ: ½ 2 : 120 Š

The proof of [2.120] depends upon the following useful relationship:

UfðAÞU
^1 ¼fUAU
^1



, ½ 2 : 121 Š

in which f(A) is an arbitrary function acting on the operator A.
Equation [2.121] can be verified by expanding f(UAU–1) as a Taylor
series. Using [2.121],

RðÞ¼ , RzðÞRyðÞRzðÞ R
y^1 ðÞR
z^1 ðÞ

¼RzðÞ exp½
i RyðÞIzR
y^1 ðފ R
z^1 ðÞ

¼RzðÞ exp½
i ðފIzcosþIxsin R
z^1 ðÞ

¼exp½
i RzðÞðÞIzcosþIxsinR
z^1 ðފ

¼exp½
i ðIzcosþIxcossinþIysinsinފ

¼exp½
i nIŠ, ½ 2 : 122 Š

which completes the desired proof. Thus, the operator for rotation about
an arbitrary angle can be represented as a series of rotations about the
yandzaxes. The five rotations used to representR( ,) in [2.120] are
not mutually independent; the rotationR( ,) can be reduced to three
independent rotations using the Euler decomposition of the general
three-dimensional rotation ( 8 ).

2.4 Quantum Mechanical NMR Spectroscopy

Theoretical analysis of an NMR experiment requires calculation
of the signal observed following a sequence of rf pulses and delays. The
initial state of the spin system is described by theequilibrium density
operator. Evolution of the density operator through the sequence of
pulses and delays is calculated using the Liouville–von Neumann
equation [2.53]. The Hamiltonian consists of the appropriate spin
interaction terms that govern evolution of the density operator.

54 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY