Various Fourier transforms
Square pulse
Consider a square pulse,
The Fourier transform of this is easy to compute,
The inverse transform of this, however, is somewhat less so. The integral we wish to compute is
Start by rewriting in a more convenient form,
and considering the two integrals separately:
Looking at and , it is clear that they both have simple poles at , which makes them difficult to integrate through straightforward means. But by allowing , and may be computed by complex contour integration in the usual way. The exponential factors will cause both integrands to decay sufficiently fast to zero for or (depending on the signs of and ) such that additional integrals of and over semicircles and of radius extending over either the upper or lower half of the complex plane, will vanish as . Thus they can be added appropriately to and to form a closed integration path and hence allow the residue theorem to be applied1. First note that both and can be written in the form where both and are holomorphic and at . Thus we have
Now consider regions in between which and switch signs. If , both and are negative, so choose and to both lie in the lower half-plane. Noting the left-handed orientation of the resulting integration loops, the residue theorem gives , which in turn gives
If , both and are positive, so choose and to both lie in the upper half-plane. Similar to the previous case (the single exception being the right-handed orientation of these loops), , so
If , then is positive while is negative. Choosing in the upper half-plane and in the lower half-plane then results in and (with the relative sign being due to the opposite orientations of the two loops), yielding
Thus we recover in its original form,
Gaussian wave packets
Consider a localized matter wave packet in the form of a Gaussian:
We wish to verify that the Fourier transform of this real-space wavefunction is also in the form of a Gaussian. Here we use the (quantum) physicist’s normalization convention:
This can easily be seen to satisfy the proper wavefunction normalization condition.
The inverse Fourier transform can be performed in a nearly identical manner,
which verifies that found above is indeed the correct Fourier transform of .
It can also be verified (by inspection, of course) that
and so the Gaussian wave packet is a minimum-uncertainty wavefunction,
We can also consider the time evolution of a Gaussian wave packet in free space. Here, our Hamiltonian is simply the kinetic energy of the packet,
Considering this, we’ll start in momentum space where is diagonal:
The time evolution of this state is found in the usual way,
and we wish to calculate its projection onto the real-space basis, . Recalling that
the integral we need to compute, then, is
Now we’ll marshal the argument of the exponential into the form with
For general , this integral can be evaluated rather easily:
So then the original integral evaluates to
And then by the Sympy theorem, we can calculate the corresponding probability density as
where we can note that the center of this wave packet travels at the group velocity , and at we recover our initial real-space wavefunction.
Footnotes
- ↵ Strictly, we must also replace the integration path close to with a small semicircle of radius and orientation opposite to that of the larger one (such that the closed path contains the pole at and take to properly apply the residue theorem.