QMGreensite_merged

Now, using [2.33],

hiA ¼

X

nm

cmcnhjmAjin: ½ 2 : 39 Š

In contrast to [2.21], the ketsjin are not necessarily the eigenfunctions
ofA; therefore, the scalar productshjmAjin do not necessarily vanish for
m 6 ¼n. Equation [2.21] is a special case derived from [2.39] if the ketsjin
are eigenfunctions ofA. For a given basis set, the termshjmAjin are
constants, and the value of the observableAfor a particular state of the
system is determined by the products of the coefficientscmcn. Once the
coefficientscmcnare known, the expectation value of any observable
can be calculated. The termAmn¼hjmAjin is the (mn)th element of the
NNmatrix representation of the operatorAin a given basis. The
productscmcncan be regarded as the elements of a matrix representation
of an operatorPdefined by

Pnm¼hjnPjim ¼cmcn: ½ 2 : 40 Š

Note that P can be explicitly written as a projection operator,
P¼ji
hj
. Substituting [2.40] into [2.39] yields

hiA ¼

X

nm

cncmhjmAjin

¼

X

nm

hjnPjimhjmAjin ¼

X

n

hjnPAjin

¼

X

nm

PnmAmn¼

X

n

ðÞPAnn

¼TrfgPA, ½ 2 : 41 Š

where Tr{} is thetraceof a matrix defined as the sum of the diagonal
elements of the matrix. The equality on line 2 of [2.41] is a consequence of
the Closure Theorem [2.37]; the equality on line 3 results from the
definition of matrix multiplication of the matrix representations of the
operators. Equation [2.41] states that the expectation value of some
observable of a system, say, for example, the amount ofx-magnetization,
is calculated as the trace of the product ofPandA.Pis the operator that is
defined by the coefficientscmcnand so describes the state of the system at
any particular point in time, andAis the operator corresponding to the
required observable. For the sake of completeness and formality,Pis a
Hermitian operator such that

hjnPjim ¼hjmPjin: ½ 2 : 42 Š

2.2 THEDENSITYMATRIX 39