section_path: Role and characterizations in mathematics/Fourier transform and Heisenberg uncertainty principle
section_name: Fourier transform and Heisenberg uncertainty principle
paragraph_text: Fourier transform and Heisenberg uncertainty principle
  
  The constant π also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as: f ^ (ξ) = ∫ − ∞ ∞ f (x) e − 2 π i x ξ d x. {\displaystyle {\hat {f}}(\xi)=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx.}
  
  Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve π somewhere. The above is the most canonical definition, however, giving the unique unitary operator on L that is also an algebra homomorphism of L to L.
  
  The Heisenberg uncertainty principle also contains the number π. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform, (∫ − ∞ ∞ x 2 | f (x) | 2 d x) (∫ − ∞ ∞ ξ 2 | f ^ (ξ) | 2 d ξ) ≥ (1 4 π ∫ − ∞ ∞ | f (x) | 2 d x) 2. {\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi)|^{2}\,d\xi \right)\geq \left({\frac {1}{4\pi }}\int _{-\infty }^{\infty }|f(x)|^{2}\,dx\right)^{2}.}
  
  The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of π in the formulae of Fourier analysis is ultimately a consequence of the Stone–von Neumann theorem, asserting the uniqueness of the Schrödinger representation of the Heisenberg group.