(May 10, 2008)
Uncertainty principles in Fourier analysis Paul Garrett
[email protected] http://www.math.umn.edu/˜garrett/
The Heisenberg Uncertainty Principle is a theorem about Fourier transforms, once we grant a certain model of quantum mechanics. That is, there is an unavoidable mathematical mechanism that yields an inequality, which has an interpretation in physics. [1] For suitable f on
R,
|f |2L2 =
Z
R
|f |2 = −
Z
R
x(f · f )0 = −2Re
Z
R
xf f
0
(integrating by parts)
That is, Z Z Z 2 0 0 2 = |f |L2 = |f | = −2Re |xf f | xf f ≤ 2
|f |2L2
R
R
R
Next, Z 2
R
0
|xf · f | ≤ 2 · |xf |L2 · |f 0 |L2
(Cauchy-Schwarz-Bunyakowsky)
Since Fourier transform is an isometry, and since Fourier transform converts derivatives to multiplications, |f 0 |L2 = |fb0 |L2 = 2π|ξ fb|L2 Thus, we obtain the Heisenberg inequality |f |2L2 ≤ 4π · |xf |L2 · |ξ fb|L2 More generally, a similar argument gives, for any xo ∈
R and any ξo ∈ R,
|f |2L2 ≤ 4π · |(x − xo )f |L2 · |(ξ − ξo )fb|L2 Imagining that f (x) is the probability that a particle’s position is x, and fb(ξ) is the probability that its momentum is ξ, Heisenberg’s inequality gives a lower bound on how spread out these two probability distributions must be. The physical assumption is that position and momentum are related by Fourier transform.
[1] I think I first saw Heisenberg’s Uncertainty Principle presented directly as a theorem about Fourier transforms
in Folland’s 1983 Tata Lectures on PDE.
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