The wave functions
capable of representing a given quantum system
belong to a function space which should be specified precisely. In order that the probability distribution of position
and momentum
have meaning, it is necessary and sufficient that the normalization condition
![]() |
(1.1) |
could be applied to the wave function
. We are thus led to the following definition of wave function space:
Where
denotes the volume
element
. Also, note that the Fourier transform
of such a function always exists: it is a square integrable function possessing the same normalization as
.
We could restrict the function space somewhat more by requiring the wave functions to be normalized to unity (eq. 1). Hwever, it turns out to be more convenient to relax this normalization condition; this can be done, as we shall see below, at the price of a slight modification in the definition of the statistical distributions and probabilities.
In the langage of mathematics, the function space defined above is a Hilbet space. It possesses indeed the properties characteristic of such a space, as shown below.
In the first place, it is a linear space. If
and
are two square integrable functions, their sum, the product of each by a complex number and, more enerally, any linear combinations
where
and
are arbitrarily chosen complex numbers, are also square-integrable functions.
In the second place, one can define a scalar product
in that space. By definition, the scalar product of the function
by the function
is
![]() |
(1.2) |
If it is zero, the functions
and
are said t be orthogonal. The norm
of a funcion
is the scalar product of this function by itself:
The fundamental properties of the scalar product are as follows:
a) the scalar product of
and
is the complex conjugate of the scalar product of
by
, namely
![]() |
(1.3) |
b) the scalar product of
by
is linear with respect to
, in other words
![]() |
(1.4) |
c) the norm of a function
is a real, non-negative number:
![]() |
(1.5) |
and if
, we have necessarily
.
All the above properties are easily deduced from the very definition of the scalar product itself. From properties (a) and (b) one easily shows that the scalar product
does not depend linearly, but antilinearly on
:
![]() |
(1.6) |
From the properties (a),(b), and (c) follows a very important property of the scalar produt, the Schwarz inequality
![]() |
(1.7) |
Equality obtains when the functions
and
are multiples of each other, and only in that case. The Schwarz inequality insures that the integral (2) defining the scalar product convertges when the functions
and
are square integrable functions.
In addition to the fact that it is linear and that one can define a scalar product there, the space of square integrable functions possesses the property of being complete; this is what allows us to identify it as a Hilbert space. To be complete means that any set of suare integrable functions satisfying the Cauchy criterion, converges (in the quadratic mean) toward a square integrable function. Conversely, any square integrable function can be considered as the limit (in the quadratic mean) of a converging series( in the sense of Cauchy) of square integrable functions (separability).
[1] Messiah, Albert. "Quantum mechanics: Volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
As of this snapshot date, this entry was owned by bloftin.