Download Uncertainty principles in Fourier analysis

(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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