Original URL is http://jcbmac.chem.brown.edu/baird/QuantumPDF/Tan_on_Hilbert_Space.html
NOTES ON HILBERT SPACE
Department of Physics
Brown University
DEFINITION: A Hilbert space is an inner product space which, as a metric space, is complete.
We will not present an exhaustive "mathematical" discussion of this subject. Rather, by using examples and analogies, hopefully you will feel more "at ease" with "Hilbert space" at the end of this short discussion.
REMARKS:
(1) Consider the space of functions, 
, where f is a "square-integrable"
complex-valued function on the real interval [a,b], ie, 
This space will be denoted as
.
One can directly verify that is a complex vector space, ie, 
, etc. We will return to clarify
in what sense one can "visualize a function as a vector".
(2) By introducing a "dot-product", 
this complex vector space becomes
an inner product space. This inner product provides us with a positive definite
"norm" for each vector, 
(3) Define a "distance" between two functions by 
. This turns into a metric space. We
shall explain the fact that is a Hilbert space next.
COMMENTS:
(1) In dealing with real numbers, it is natural to first start working with integers.
One then finds that one needs rationals, and finally irrationals. To formally introduce
"irrationals", one can first introduce a notion of distance, d(x,y),
between two numbers x and y, which of course can be chosen to be the
absolute value 
. (With a notion of distance, the space of numbers becomes a "metric
space".) Now one can talk about an infinite sequence of numbers, e.g.,
{3, 3.1, 3.14, 3.141, 3.1415, . . .}. Note that (i) the differences between any pairs
of numbers further down the sequence become smaller and smaller. (This is an example
of a "Cauchy sequence".) (ii) As finite decimals, every term in the sequence
is a rational number. Now, as you can guess, the limit of this sequence is 
, and it is not a rational
number.
(2) A metric space is not complete if it contains Cauchy sequences whose
limit points are not contained in the space. As the example above shows, the space
of rational numbers, with the usual notion of distance, is not a complete
metric space. By including all irrationals, the space of all real numbers is complete.
Mathematically, one states that the real numbers, 
, is the completion of the
set of rational numbers.
(3) We all agree "irrational numbers exist" and they must be included
in any sensible usage of numbers. However, for physics, it is also clear that, for
a given problem, it is sufficient to work with rationals only, e.g. by agreeing
to always work to the twentieth decimals. Another way of stating this fact is that
"one can always approximate an irrational number by a rational number to any
degree of accuracy one desires." Mathematically, one states that the set of
rational numbers is dense in .
(4) It is a mathematical fact that the space , defined by Eq. I1, as a metric
space with a "distance between functions f and g" defined by , is complete. That is,
the "limiting function" of any Cauchy sequence of functions in is also in . Therefore, is a Hilbert space, which
will also be denoted by 
. [Note: The notation 
will be used to denote the Hilbert
space of functions defined over 
].
(5) Let us clarify the situation by drawing an analogy with the case of real numbers.
The role of rationals is now played by continuous functions on the interval [a,b].
However, the condition of "square-integrability", I1, is much less stringent
than continuity. Many functions which are not continuous nevertheless satisfy I1.
It can be shown that these "additional" functions can always be thought
of as limits of Cauchy sequences of continuous functions. We "complete"
by adding to the set of continuous functions these "limiting points". That
is:
is the completion of the space of continuous functions on the interval [a,b],
with respect to a distance defined by .
(6) Just like the situation with real numbers, any function 
can always be approximated by continuous
functions, to any desired degree of accuracy. Mathematically, one states that the
set of continuous functions on [a,b] is dense in .
(7) Let us end with an example using Fourier series. let us consider defined over the interval 
.
Consider a periodic function f(x). Assume that f(x) is continuous and differentiable
except at a point 
, where f(x) has a finite discontinuity. Let us consider the following
sequence of functions, 
, defined by
where
for all integers n. Note that each 
is continuous at 
where f(x) has a discontinuity.
For any small number 
, we can always choose N large enough so that
for N' >= N. That is, to the accuracy , f can be approximated by a continuous function,
,
although f(x) is not.
The set of 
is a Cauchy sequence. Therefore, the space of continuous functions is
not complete. However, the space is. Indeed, it is precisely in the sense of
I.3 that we understand the statement that every periodic function of period 2 can
be represented by a Fourier series: 
with 
given by (I.2).
DEFINITION: A system of functions, 
, defined on [a,b] is said to be
an orthonormal system if 
DEFINITION: A system of functions, , defined on [a,b] and belonging
to ,
is said to be complete if there exists no functions different from zero in
which is orthogonal to all functions 
.
Without a loss of generality, let us now shift the interval [a,b] to [-L/2, L/2].
Consider the set of basis functions, 
, 
, defined by
We have previously pointed out that this set forms an orthonormal set:
For each function F in , consider the Fourier series representation
The fact that any function in can be expanded in this basis can be used
to show that this set is complete. By substituting (II.5) into (II.4), one
finds that the completeness can be expressed mathematically as 
COMMENTS:
(1) This set of complete orthonormal basis functions will be referred to
as our standard basis. In terms of this standard basis, every function F
can be expanded as in (II.4). Instead of writing out (II.4) explicitly, we can also
represent the function F by a "column vector", with components:
.
(To save space, let's not write it out as a "column" here.) That is, in
this basis, we can associate a function with a vector with infinite many "components".
In this representation, one "sees" a function as a column vector, just
like the situation with a finite dimensional vector space.
(2) To draw a closer analogy with our "standard representation" for
a finite dimensional vector space, let's re-write the inner-product, (II.2), and
the norm, (II.3), as follows: Consider two functions, F(x) and F'(x),
described by components 
and 
respectively in this standard basis. It follows
from (II.3) that
(3) In this representation, the "vector" nature of the space is manifest! Instead of representing
a vector by its "components", one can also represent a function, F(x),
by an abstract vector in a Dirac notation, 
. If we first introduce a Dirac
notation for our standard basis vectors: 
, , Eq. (II.4) can be written as:
with
The orthonormality condition in a Dirac notation becomes
We will shortly show that the completeness condition in a Dirac notation is
You should now compare these representations with those for a finite dimensional vector space, and convince yourself that these two sets are formally identical.
(4) You should also convince yourself that every linear operator on 
,
can be associated with an infinite by infinite matrix as follows:
What is the form of the matrix 
for the momentum operator 
?
"images/tan It is convenient to formally define a set of basis
vectors, 
, labelled by a continuous index, 
, satisfying Dirac-delta
orthonormality condition as follows: "images/tan
In terms of this basis, it is now possible to express each abstract vector 
in
as
Coordinate Representation: 
It follows from (III.1) and (III.2) that f(x) can be recovered by "dotting"
a bra vector 
into Eq. (III>2), ie,
That is, we have achieved a representation where the original function f(x)
in
now appears as "components" of the abstract vector in a coordinate basis. In particular,
the inner product of two vectors, and 
, through (III.3) and (III.2), is
in agreement with the original definition, (III.2).
Since every vector in can be expressed by (III.2), they form a complete basis:
Completeness: 
COMMENTS:
(1) We claim that 
is an eigenstate of our position operator, 
,
ie,
for . Eq. (III.6) follows if one writes down the components of , ie,
That is, 
is the coordinate wavefunction for the eigenvector of with an eigenvalue 
. By applying (III.3) to (III.7),
one simply reproduces the orthogonality condition, (III.1).
(2) In this representation, every linear operator can be expressed as
This representation follows by considering 
and use (III.5) twice. For instance,
Whenever 
is proportional to , the operator 
will be called a local operator. Instead
of a "double-sum", Eq. (III.9), one has a single-sum representation:
For local operators, one often simply writes down the matrix for the operator ,
Better yet, one can drop the factor entirely
if it is understood that one is dealing with a local operator! For instance, instead of writing out
explicitly, it is much simpler to accept the notation
[Convince yourself that the Hamiltonian operator 
is local in a coordinate representation,
and make sure you understand the proper interpretation of the expression 
]
(3) Lastly, let's return to our "standard basis", . It is clear that the abstract
vector, 
, is given by
With this understanding, by sandwiching (II.12) between 
and 
, it leads to (II.6), which proves
(II.12), as promised.