14 - The Uncertainty Principle

Summary

To get some feeling for the size of the limit given by Heisenberg's relation, suppose that the position of an electron is measured to the accuracy of one part in a nanometer (10₋₉ m); then the momentum would become so uncertain that one could not expect that, one second later, the electron would be closer than 100 kilometers away!

R. Penrose [1989, p. 248]

The uncertainty principle is widely known for its “philosophical” applications: in quantum mechanics, of course, it shows that a particle's position and momentum cannot be determined simultaneously (Heisenberg [1930]); in signal processing it establishes limits on the extent to which the “instantaneous frequency” of a signal can be measured (Gabor [1946]). However, it also has technical applications, for example in the theory of partial differential equations (Fefferman [1983]).

D. Donoho and P. Stark [1989, p. 906]

The classical uncertainty principle says that a function and its Fourier transform cannot both be highly localized or concentrated. In quantum mechanics, this becomes the statement that it is impossible to find a particle's position and momentum simultaneously, as Penrose makes quite explicit in the quote above. But, as Donoho and Stark remark, the uncertainty principle shows its (ugly/beautiful) face in many other areas such as signal processing and medical imaging. References for this chapter are Donoho and Stark [1989], Dym and McKean [1972], Grünbaum [1990], Smith [1990], Terras [1985, Vol. I], Elinor Velasquez [1998], and Wolf [1994].