Heisenberg’s uncertainty principle says there is a limit to how well you can know both the position and momentum of anything at the same time. The product of the uncertainties in the two quantities has a lower bound.

There is a closely related principle in Fourier analysis that says a function and its Fourier transform cannot both be localized. The more concentrated a signal is in the time domain, the more spread out it is in the frequency domain.

There are many ways to quantify how localized or spread out a function is, and corresponding uncertainty theorems. This post will look at the form closest to the physical uncertainty principle of Heisenberg, measuring the uncertainty of a function in terms of its **variance**. The Fourier uncertainty principle gives a lower bound on the product of the variance of a function and the variance of its Fourier transform.

## Variance

When you read “variance” above you might immediately thing of variance as in the variance of a random variable. Variance in Fourier analysis is related to variance in probability, but there’s a twist.

If *f* is a real-valued function of a real variable, its variance is defined to be

This is the variance of a random variable with mean 0 and probability density |*f*(*x*)|². The twist alluded to above is that *f* is not a probability density, but |*f*(*x*)|² is.

Since we said *f* is a real-valued function, we could leave out the absolute value signs and speak of *f*(*x*)² being a probability density. In quantum mechanics applications, however, *f* is complex-valued and |*f*(*x*)|² is a probability density. In other words, we multiply *f* by its complex conjugate, not by itself.

The Fourier variance defined above applies to any *f* for which the integral converges. It is not limited to the case of |*f*(*x*)|² being a probability density, i.e. when |*f*(*x*)|² integrates to 1.

## Uncertainty principle

The Fourier uncertainty principle is the inequality

where the constant *C* depends on your convention for defining the Fourier transform [1]. Here ||*f*||_{2}² is the integral of *f*², the square of the *L*² norm.

Perhaps a better way to write the inequality is

for non-zero *f*. Rather than look at the variances per se, we look at the variances relative to the size of *f*. This form is scale invariant: if we multiply *f* by a constant, the numerators and denominators are multiplied by that constant squared.

The inequality is exact when *f* is proportional to a Gaussian probability density. And in this case the uncertainty is easy to interpret. If *f* is proportional to the density of a normal distribution with standard deviation σ, then its Fourier transform is proportional to the density of a normal distribution with standard deviation 1/σ, if you use the radian convention described in [1].

## Example

We will evaluate both sides of the Fourier uncertainty principle with

h[x_] := 1/(x^4 + 1)

and its Fourier transform

g[w_] := FourierTransform[h[x], x, w]

We compute the variances and the squared norms with

v0 = Integrate[x^2 h[x]^2, {x, -Infinity, Infinity}]
v1 = Integrate[w^2 g[w]^2, {w, -Infinity, Infinity}]
n0 = Integrate[ h[x]^2, {x, -Infinity, Infinity}]
n1 = Integrate[ g[w]^2, {w, -Infinity, Infinity}]

The results in mathematical notation are

From here we can calculate that the ratio of the left side to the right side of the uncertainty principle is 5/18, which is larger than the lower bound *C* = 1/4.

By the way, it’s not a coincidence that *h* and its Fourier transform have the same norm. That’s always the case. Or rather, that is always the case with the Fourier convention we are using here. In general, the *L*² norm of the Fourier transform is proportional to the *L*² norm of the function, where the proportionality constant depends on your definition of Fourier transform but not on the function.

Here is a page that lists the basic theorems of Fourier transforms under a variety of conventions.

## Related posts

[1] In the notation of the previous post *C* = 1/(4*b*²). That is, *C* = 1/4 if your definition has a term exp(± *i* ω *t*) and *C* = 1/16π² if your definition has a exp(±2 π *i* ω *t*) term.

Said another way, if you express frequency in radians, as is common in pure math, *C* = 1/4. But if you express frequency in Hertz, as is common in signal processing, *C* = 1/16π².

Fourier transform definitions also have varying constants outside the integral, e.g. possibly dividing by √2π, but this factor effects both sides of the Fourier uncertainty principle equally.