Several weeks ago I wrote about the Fourier uncertainty principle which gives a lower bound on the product of the variance of a function *f* and the variance of its Fourier transform. This post expands on the earlier post by quoting some results from a recent paper [1].

## Gaussian density

The earlier post said that the inequality in the Fourier uncertainty principle is exact when *f* is proportional to a Gaussian probability density. G. H. Hardy proved this result in 1933 in the form of the following theorem.

Let *f* be a square-integrable function on the real line and assume *f* and its Fourier transform satisfy the following bounds

for some constant *C*. Then if *ab* > 1/4, then *f* = 0. And if *ab* = 1/4, *f*(*x*) = *c* exp(-*ax*²) for some constant *c*.

Let’s translate this into probability terms by setting

Now Hardy’s theorem says that if *f* is bounded by a multiple of a Gaussian density with variance σ² and its Fourier transform is bounded by a multiple of a Gaussian density with variance τ², then the product of the two variances is no greater than 1. And if the product of the variances equals 1, then *f* is a multiple of a Gaussian density with variance σ².

## Heisenberg uncertainty

Theorem 3 in [1] says that if *u*(*t*, *x*) is a solution to the free Schrödinger’s equation

then *u* at different points in time satisfies a theorem similar to Hardy’s theorem. In fact, the authors show that this theorem is equivalent to Hardy’s theorem.

Specifically, if *u* is a sufficiently smooth solution and

then αβ > (4*T*)^{-2} implies *u*(*t*, *x*) = 0, and αβ = (4*T*)^{-2} implies

## Related posts

[1] Aingeru Fernández-Bertolin and Eugenia Malinnikova. Dynamical versions of Hardy’s uncertainty principle: A survey. Bulletin of the American Mathematical Society. DOI: https://doi.org/10.1090/bull/1729