##

Building a Localized Single-Particle Wave Packet

We now have a wave function for a
**free particle with a definite momentum**

where the wave number
is defined by
and the angular frequency
satisfies
.
It is not localized since
everywhere.
We would like a state which is localized and **normalized to one particle**.

To make a **wave packet which is localized in space**,
we must add components of different wave number.
Recall that we can use a
Fourier Series
to compose any function
when we limit the range to
.
We do not want to limit our states in
, so we will take the limit that
.
In that limit, every wave number is allowed so the sum turns into an integral.
The result is the very closely related
Fourier Transform

with coefficients which are computable,

The normalizations of
and
are the same
(with this **symmetric form**)
and both can represent probability amplitudes.

We understand
as a **wave packet** made up of definite momentum terms
.
The coefficient of each term is
.
The probability for a particle to be found in a region
around some value of
is
.
The probability for a particle to have wave number in region
around some value of
is
.
(Remember that
so the momentum distribution is very closely related.
We work with
for a while for economy of notation.)

Jim Branson
2013-04-22