QMGreensite_merged

Using the Pauli spin matrices, [2.135] can be written as

ðÞ¼t UðÞ 0 U
^1 ¼
expðÞ
i tIzIyexpðÞ¼i tIz IxsinðÞ
t IycosðÞ t:

½ 2 : 136 Š

Note that att¼0,(t)¼
Iyand at t¼/2,(t)¼Ix. Magnetization
with a positive resonance offset in the rotating frame precesses in the
sensex!y!
x!
y.
The preceding results yield the form of the detectable magnetization
for a one-pulse sequence for a single isolated spin:

Mþ

ðtÞ¼

N h^2! 0

2 kBT

Tr Ixsinð tÞ
Iycosð tÞ

   

IxþiIy



¼

N h^2! 0

2 kBT



TrI^2 x



sinð tÞþiTrIxIy



sinð tÞ

TrIyIx



cosð tÞ
iTr



I^2 y



cosð tÞ

¼

N h^2! 0

8 kBT

sinð tÞ
icosð tÞ

   

¼

N h^2! 0

8 kBT

expið t
= 2 Þ

   

, ½ 2 : 137 Š

in which all constants have been reintroduced. This signal has the form
expected from the analysis of the same system using the Bloch model
in the absence of relaxation (the factor exp[–i/2] is a time-independent
phase factor).

2.5 Quantum Mechanics of Multispin Systems

In this section, the use of the density operator approach to perform
calculations on larger, scalar coupled spin systems will be illustrated;
as discussed in Chapter 1, Section 1.6, the Bloch model fails to properly
account for the evolution of such spin systems. The problem is to
establish the matrix representation of wavefunctions and operators in a
two-spin (in general,N-spin) system and derive an appropriate operator
algebra. The central results will be derived using the direct product basis;
transformations to other basis sets can be performed using similarity
transformations as described previously. Additional details can be found
in the monograph by Corio ( 8 ).

58 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY