As we have discussed in lecture, there is a close relationship between the measurement of position and momentum and the mathematical concept of the Fourier Transform. Many of the known mathematical properties of the Fourier Transform correspond directly to physical concepts which we have only yet been able to discuss qualitatively. To be able to discuss these physical concepts on a quantitative level, we must develop a working knowledge of the Fourier transform. The purpose of this problem is to develop just such knowledge.
The operation of the Fourier transform we shall write as
The inverse relationship is
For this problem you will consider a wave packet where
a) Compute the value N such that 
is properly
normalized and use this value for the rest of the problem.
b) Compute 
from 
using the definition
(1) and verify that 
is properly normalized so
long as 
is properly normalized.
c) Compute 
from your expression for 
using the definition (2) and verify that you recover
(3).
d) Sketch the real part of both 
and 
,
indicating in your sketches, in terms of the parameters 
, 
and D, the approximate width and center of each function and
the wavelength of any oscillations.
e) Confirm that your results satisfy the general Fourier theorem
on the scaling of functions: if 
then 
. State what this means in terms of the behavior
of the functions 
and 
as the parameter s
varies. Interpret your result physically in terms of the uncertainty
principle.
f) Confirm that your results satisfy the general Fourier theorem
on the displacement of functions: if 
then 
. Interpret
this result physically in terms of the effect of this operation on the
momentum distribution and on the oscillatory behavior of the packet in
real space.
g) Confirm that your results satisfy the general Fourier theorem
on the multiplication by a phase of functions: if 
then 
. How is
a shift in the position of the center of the wave packet reflected in
the momentum wavefunction 
? How is it reflected in the
momentum probability distribution 
?
h) Confirm, for the case n=1, that your results
satisfy the general Fourier theorem on the moment-derivative
relationship: if 
then 
.
Give an
interpretation of this result in terms of the change
which
undergoes under an infinitesimal displacement 
, 
and the corresponding operations on 
.
i) Compute <X>, 
,
and 
for this wave packet. Confirm
that the uncertainty principle 
holds
in this case.
You may find the following integrals useful:
