Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Fourier Transforms in Quantum Physics

Fourier transforms are used in many branches of physics and engineering to relate functions described in terms of conjugate variables. These variables are those such as wavevector and position, or frequency and time, which are associated with waves and particles respectively and thereby obey a version of the uncertainty principle. In quantum physics these relations are extended to position and momentum, or energy and time, via the introduction of Planck's constant h in p = (h/2p) k and E = hn.  

The first search of the web that we did under (Fourier + transform) gave 8358 entries, more than we really need for this course.... A list of the more useful sites we have found to date is given below: you could also go to the library or get out your old math books, but this is more fun.

Everyone is clear that the definition of the one dimensional (1D) transform g(k) into f(x) is defined as 
  

and that the inverse transform is  
  

but the problem arises if you are picky about the position of the 2 p. The symmetric form has 1/root(2p ) in both equations. The other possibility is to have the 2p in the exponent, as is used for example in Cowley's Diffraction Physics, chapter 2. Then one doesn't need anything up front.

Much of the time we are only interested in shapes of the functions and in this case the 2 p's, or in general the normalization constants, don't matter.

A related technique used to describe diffraction paterns is the Convolution theorem. This is described by Nimish Hathi, using the example of the familiar two-slit diffraction pattern.  

Some of the most obvious 1D transforms are
  • f(x) = top hat function of width a, g(k) = sinX/X; what is X?
  • f(x) = Gaussian with standard deviation s, g(k) = ? with width ? Try this first, and then see if you agree with Violet Taylor's solution.
  • f(x) = Aexp(-a |x|), g(k) = ? See if you can do this, and compare your answer with Kevin Healy's solution.
  • f(x) = Lorentzian A/(x2 + a2), g(k) = ? Try to remember what a contour integral looks like, and compare your answer with Mike Grams' solution.
Get back to us with derivations and pictures of useful functions and we will link your solutions into this page. More exotic, but still useful links are 

Latest version of this document: 25th January 2005.