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
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
The normalizations of
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