5
$\begingroup$

The uncertainty principle (UP) comes up in engineering and physics, but it is a mathematical idea. An old text describes it as "reciprocal spreading." If $f$ is a well-behaved function, the UP might be expressed as $W(f)W(\hat{f}) \geq k$, where $k$ is some constant. If $g$ is a Gaussian, we get equality, i.e., $W(g)W(\hat{g}) = k$.

My question is this. At least in Fourier analysis, the Gaussian is sort of a minimum in the above sense. Are there any real-world problems for which this is a solution? Even in EE I don't think "optimality" of the Gaussian with respect to the UP is ever used.

Thanks for any thoughts.

  • 0
    I'm not sure but thanks for the ref and I will look at it.2011-10-27

1 Answers 1

1

Assuming you are looking for a "real-world" application of the mininum-uncertainty property of the Gaussian, I might have one answer:

In quantum mechanics, Gaussians are used to create minimum-uncertainty wavefunctions which are solutions of the Schrödinger equation. The minimum-uncertainty solutions are useful in constructing what are known as coherent states.

See here:

http://en.wikipedia.org/wiki/Coherent_states

  • 0
    I do not pretend to completely understand the physics but clearly you are right, "minimal uncertainty" (in the sense of my question) is associated with the coherent states described in the article, so this is right on point. Thanks!2012-06-22