uncertainty theorem

The uncertainty principle, as first formulated by Heisenberg, states that the product of the standard deviations of two conjugated variablesMathworldPlanetmath, cannot be less than some minimum. This statement has been generalized to a precise mathematical theorem, in the frame of waveletMathworldPlanetmath theory.


Let f(t) be a real function of the real variable, satisfying the L2 condition (see below), and F(ω) its Fourier transformDlmfMathworldPlanetmath. The standard deviation Δt and Δω of t and ω respectively, satisfy the following inequality:


For this formula to make sense, Δt and Δω must be precisely defined.


A real function f(t) of the real variable t will be said to satisfy the L2 condition if f(t), tf(t) and the derivative f(t) are all in L2.
If F(ω) is its Fourier transform, -iωF(ω) is the transform of f(t). All the following functionsMathworldPlanetmath belong to L2:


The first three functions are just the definition and the two last ones result from Parseval’s identityPlanetmathPlanetmath, recalled here in its integral form:


U(ω) and V(ω) are the Fourier transforms of u(t) and v(t).


f(t) and F(ω) being in L2, we may define their finite norms:

f2=-f(t)2𝑑t   F2=-F(ω)2𝑑ω

By Parseval’s identity, they are related:


We are now able to define the probability distributions T(t) and Ω(ω) for the ”random” variables t and ω:

T(t)=f(t)2f2   Ω(ω)=F(ω)2F2

Since the L2 integrals of T and Ω are 1, they are proper probability distributions. The mean value t0 of t is defined the usual way:


Note that ω’s mean value is always 0 because f(t) is a real function. Finally, we have the standard deviations for the uncertainty theorem:

Δt2=-T(t)(t-t0)2𝑑t   Δω2=-Ω(ω)(t-t0)2𝑑ω


The heart of the proof is the Cauchy-Schwarz inequality in the L2 Hilbert spaceMathworldPlanetmath: the product of the norms of two functions u(t) and v(t) is greater than, or equal to, the norm of their scalar productMathworldPlanetmath:


Equality occurs if, and only if, one of the functions is proportional to the other. For the two functions u(t)=(t-t0)f(t) and v(t)=f(t), we have therefore:


The integral at the right hand side can be integrated by parts. Using the definition of f:


But f and F are related by a 2π factor, so:


Applying Parseval’s identity to the second integral of the left hand side, we get:


We have used the fact that the Fourier transform of f(t) if -iωF(ω).
Now, dividing both sides by the norms, and simplifying by the 2π factor, we get exactly the uncertainty theorem:



The Cauchy-Schwarz inequality becomes an equality if, and only if, one of the functions is proportional to the other. In our case, this condition is expressed by f(t)=λ(t-t0)f(t) where λ is a constant. This differential equation is readily solved: f(t)=keλ(t-t0)2. f(t) must be in L2 so that λ must be negative. Defining λ=-12σ2, we get the gaussian function in its traditional form:


The constant k has been omitted because it cancels anyway in the probability distributions. The standard deviations are easily computed from their definitions:

Δt=σ2    Δω=1σ2

Their product is 12, independent of σ. There is no other function with this property.


  • 1 Roberto Celi Time-Frequency visualization of helicopter noise
    Despite its frightening title, this paper is mostly theoretical and it is the only place where I saw the uncertainty theorem clearly stated.
  • 2 Robi Polikar The wavelet tutorial
    http://users.rowan.edu/ polikar/wavelets/wttutorial.html
Title uncertainty theorem
Canonical name UncertaintyTheorem
Date of creation 2013-03-22 18:50:37
Last modified on 2013-03-22 18:50:37
Owner dh2718 (16929)
Last modified by dh2718 (16929)
Numerical id 14
Author dh2718 (16929)
Entry type Theorem
Classification msc 42C40
Related topic UncertaintyPrinciple