Next: Slater Determinants
Up: intro_estruc
Previous: Postulates of Quantum Mechanics
For the purposes of solving the electronic Schrödinger equation on a
computer, it is very convenient to turn everything into linear algebra.
We can represent the wavefunctions as vectors:

(5) 
where
is called a ``state vector,'' are the expansion
coefficients (which may be complex), and
are
fixed ``basis''
vectors. A suitable (infinitedimensional) linear vector space for such
decompositions is called a ``Hilbert space.''
We can write the set of coefficients
as a column
vector,

(6) 
In Dirac's ``bracket'' (or braket) notation, we call
a ``ket.''
What about the adjoint of this vector? It is a row vector denoted
by
, which is called a ``bra'' (to spell ``braket''),

(7) 
In linear algebra, the scalar product
between two vectors
and is just

(8) 
assuming the two vectors are represented in the same basis set and that the
basis vectors are orthonormal (otherwise, overlaps between the basis
vectors, i.e., the ``metric,'' must be included). The quantum mechanical
shorthand for the above scalar product in braket notation is just

(9) 
Frequently, one only writes the subscripts and in the Dirac
notation, so that the above dot product might be referred to as
just
. The order of the vectors and
in a dot product matters if the vectors can have complex numbers
for their components, since
.
Now suppose that we want our basis set to be every possible value of
coordinate . Apart from giving us a continuous (and infinite) basis
set, there is no formal difficulty with this. We can then represent an
arbitrary state as:

(10) 
What are the coefficients ? It turns out that these coefficients
are simply the value of the wavefunction at each point . That is,

(11) 
Since any two coordinates are considered orthogonal (and their overlap
gives a Dirac delta function), the scalar product of two state functions in
coordinate space becomes

(12) 
Now we turn our attention to matrix representations of operators. An
operator can be characterized by its effect on the basis
vectors. The action of on a basis vector
yields some new vector
which can be expanded in terms of the
basis vectors so long as we have a complete basis set.

(13) 
If we know the effect of on the basis vectors, then we know the
effect of on any arbitrary vector because of the linearity
of .
or

(15) 
This may be written in matrix notation as

(16) 
We can obtain the coefficients by taking the inner product of
both sides of equation 13 with ,
yielding
since
due to the orthonormality
of the basis. In braket notation, we may write

(18) 
where and denote two basis vectors. This use of braket
notation is consistent with its earlier use if we realize that
is just another vector .
It is easy to show that for a linear operator , the inner
product
for two general vectors (not
necessarily basis vectors) and is given by

(19) 
or in matrix notation

(20) 
By analogy to equation (12), we may generally write
this inner product in the form

(21) 
Previously, we noted that
, or
. Thus we can see also that

(22) 
We now define the adjoint of an operator , denoted by
, as that linear operator for which

(23) 
That is, we can make an operator act backwards into ``bra'' space
if we take it's adjoint.
With this definition, we can further see that

(24) 
or, in braket notation,

(25) 
If we pick
and
(i.e.,
if we pick two basis vectors), then we obtain
But this is precisely the condition for the elements of a matrix and its
adjoint! Thus the adjoint of the matrix representation of is
the same as the matrix representation of
.
This correspondence between operators and their matrix representations
goes quite far, although of course the specific matrix representation
depends on the choice of basis. For instance, we know from linear
algebra that if a matrix and its adjoint are the same, then the matrix
is called Hermitian. The same is true of the operators; if

(27) 
then is a Hermitian operator, and all of the special properties
of Hermitian operators apply to or its matrix representation.
Next: Slater Determinants
Up: intro_estruc
Previous: Postulates of Quantum Mechanics
David Sherrill
20030807