QMGreensite_merged

Although this expression is an infinite series of terms, a compact closed-
form solution often can be obtained if recursive relationships can be
identified after evaluating a small number of terms. For example, in
evaluating the propagator exp(
iIz)Ixexp(iIz) using the BCH formula,
the series expansion can be separated into two parts that represent
the series expansions of cosand sin, leading to the compact solution
IxcosþIysin. The BCH formula provides an alternative approach to
that outlined in Section 2.3 for determining the effects of propagators on
the density operator.

2.2.5 MATRIXREPRESENTATIONS OF THESPINOPERATORS
To proceed, the matrix representation of the angular momentum
operators that uses theji andj istates of the spin as basis functions
must be presented. As shown by [1.1], the intrinsic spin angular
momentum has units ofh. Consequently, the spin angular momentum
operators also have units ofh, as shown directly by [2.31]. However,
in the remainder of this text, the spin operators will be defined as
dimensionless quantities by mapping I!I=h. The constant of
proportionalityhwill be reintroduced explicitly as necessary. As will
be seen, a dimensionless set of spin angular momentum operators is
particularly useful for analyzing the evolution of the density operator,
which is itself dimensionless. The Pauli spin matrices form a complete
basis set for a single spin-1/2 system ( 5 ):

Ix¼

1

2

01

10



, Iy¼

1

2

0 
i

i 0



, Iz¼

1

2

10

0 
 1



: ½ 2 : 71 Š

Each of these operators is Hermitian. The spin operators satisfy the
commutation relation

Ix,Iy

   

¼iIz ½ 2 : 72 Š

and any cyclic permutation of [2.72], i.e., [Iz,Ix]¼iIyand [Iy,Iz]¼iIx.
The eigenkets are represented by the 21 column vectors,

ji ¼

1

0



,

¼

0

1



, ½ 2 : 73 Š

and the eigenbras are represented by the 12 row vectors,

hj ¼ 10

   

,

(^)
¼ 01


: ½ 2 : 74 Š
46 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY