QMGreensite_merged

Using the Dirac notation, an arbitrary wavefunction,
, can be
written as a superposition of a set of orthonormal time-independent kets,
known as eigenkets or basis kets,

ji
 ¼

XN

n¼ 1

cnjin, ½ 2 : 33 Š

wherejin are the basis kets (e.g., the and wavefunctions), cnare
complex numbers, andNis the dimensionality of the vector space. For
example, the wavefunction for a system consisting of a single spin-1/2
nucleus can be described by the linear combination of the kets for the
and states of that nucleus (which are the eigenfunctions of the angular
momentum operator). The coefficients,cn, can be regarded as amplitude
factors that describe how much a particular basis ket contributes to
the total wavefunction at any particular time. The basis kets are time
independent; consequently, any time dependence in
is contained in the
complex coefficients. Premultiplying [2.33] by the bra,hmj, and applying
the orthogonality condition yields,

cm¼himj
, ½ 2 : 34 Š

so that

ji
 ¼

XN

n¼ 1

cnjin ¼

XN

n¼ 1

hjn
ijin ¼

XN

n¼ 1

jinhjn
i: ½ 2 : 35 Š

The latter identity suggests thatjinhjn is an operator acting on
such
that

jinhinj
 ¼cnjin: ½ 2 : 36 Š

Because [2.35] must hold for arbitrary
, the useful Closure Theorem is
obtained immediately,

XN

n¼ 1

jinhjn ¼E, ½ 2 : 37 Š

in whichEis the identity operator. The operator jinhjn is called a
projection operator because it ‘‘projects out’’ from
the component
ketjin.
The expectation value of some property,hiA, can be written in Dirac
notation as

hiA ¼

Z


A
d¼hj
Aji
: ½ 2 : 38 Š

38 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY