212 Transform
Techniques
in Physics
This
is
the way we had found a representation
of
the Dirac delta
function previously. The Fourier transform approaches a constant
in
this limit.
As
a approaches zero, the sinc function approaches one, leaving
j (k)
~
2ab =
1.
Thus, the Fourier transform
of
the Dirac delta
function
is
one. Namely, we have
.c
o( x
)e
ikx
dx =
1.
In
this case
we
have that the more localized the function f (x)
is,
the more spread out the Fourier transform
is.
The Uncertainty Principle: The widths
of
the box function and its
Fourier transform are related
as
we have seen
in
the last two limiting
cases. It
is
natural to define the width, Ilx,
of
the box function as
Ilx
= 2a.
The width
of
the Fourier transform is a little trickier. This function
actually extends along the entire k-axis. However,
as j (k) becomes
more localized, the central peak becomes narrower. So,
we
define
the width
of
this function,
Me
as the distance between the first zeros
A 2b
on
either side
of
the main lobe. Since f (k) =T sin
lea,
the zeros
are at the zeros
of
the sine function, sin
lea
=
O.
The first zeros are
1t
at k = ± - . Thus,
a
21t
ilk=-.
a
Combining the expressions for the two widths, we find that
Ilxll.k =
41t.
Thus, the more localized a signal (smaller ox), the less localized its
transform (larger
ok).
This notion is referred to as the Uncertainty Principle. For more
general signals, one needs to define the effective widths more carefully,
but the main idea holds.
ilxilk
~
1.
While this is a result
of
Fourier transforms, the uncertainty principle
arises in other forms elsewhere. In particular, it appears in quantum
mechanics, where is it most known. In quantum mechanics, one finds that
the momentum
is
given
in
terms
of
the wave number, p = lik, where Ii is
Planck's constant divided by
21t.
Inserting this into the above condition,
one obtains