uncertainty principle

We will find the Fourier transformMathworldPlanetmath

F(ω):=12π-f(t)e-iωt𝑑t (1)

of the Gaussian bell-shaped function

f(t)=Ce-at2 (2)

where C and a are positive constants.

We get first


Completing the square in


and substituting  a(t+iω2a):=z,  we may write

F(ω)=C2π-e-a(t+iω2a)2e-ω24a𝑑t=C2πae-ω24ale-z2𝑑z, (3)

where l is a line of the complex planeMathworldPlanetmath parallel to the real axis and passing through the point  z=iω2a.  Now we can show that the integral


does not depend on y at all.  In fact, we have

Iyy=-ye-(x+iy)2𝑑x=-2i-e-(x+iy)2(x+iy)𝑑x=i/x=-e-(x+iy)2=i/x=-e-x2+y2e-2ixy= 0.

Hence we may evaluate Iy as


(see the area under Gaussian curve).  Putting this value to (3) yields the result

F(ω)=C2ae-ω24a. (4)

Thus, we have gotten another Gaussian bell-shaped function (4) corresponding to the given Gaussian bell-shaped function (2).

InterpretationMathworldPlanetmathPlanetmath.  One can take for the breadth of the bell the portion of the abscissa axis, outside which the ordinate drops under the maximum value divided by e, for example.  Then, for the bell (2) one writes


whence  t=1a  giving, by evenness (http://planetmath.org/EvenFunction) of the function, the breadth  Δt=2a.  Similarly, the breadth of the bell (4) is  Δω=4a.  We see that the productPlanetmathPlanetmath

ΔtΔω=8 (5)

has a constant value.  One can show that any other shape of the graphs of f and F produces a relationMathworldPlanetmath to (5).  The breadths are thus inversely proportional (http://planetmath.org/Proportion).

If t is the time and f is the on a system of oscillators with their natural frequencies, then in the formulaMathworldPlanetmath


of the inverse Fourier transform, F(ω) means the amplitude of the oscillator with angular frequency ω.  We can infer from (5) that the more localised (Δt small) the external force is in time, the more spread (Δω great) is its spectrum of frequencies, i.e. the greater is the amount of the oscillators the force has excited with roughly the same amplitude.  If one, conversely, wants to better the selectivity, i.e. to compress the spectrum narrower, then one has to spread out the external action in time.  The impossibility to simultaneously localise the action in time and enhance the selectivity of the action is one of the manifestations of the quantum-mechanical uncertainty principle, which has a fundamental role in modern physics.


  • 1 Я. Б. Зельдович &  А. Д. Мышкис: Элементы  прикладной  математики.  Издательство  ‘‘Наука’’.  Москва (1976).
  • 2 Ya. B. Zel’dovich and A. D. Myshkis: ‘‘Elements of applied mathematics’’. Nauka (Science) Publishers, Moscow (1976).
Title uncertainty principle
Canonical name UncertaintyPrinciple
Date of creation 2013-03-22 18:38:25
Last modified on 2013-03-22 18:38:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Example
Classification msc 42A38
Related topic UncertaintyTheorem
Related topic SubstitutionNotation
Related topic GraphOfEquationXyConstant