Editing Pi (section)

=== Fourier transform and Heisenberg uncertainty principle ===
[[File:Animation of Heisenberg geodesic.gif|thumb|right|An animation of a [[Heisenberg group#As a sub-Riemannian manifold|geodesic in the Heisenberg group]]]]
The constant {{pi}} also appears as a critical spectral parameter in the [[Fourier transform]].  This is the [[integral transform]], that takes a complex-valued integrable function {{math|''f''}} on the real line to the function defined as:

<math display=block>\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi}\,dx.</math>

Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve {{pi}} ''somewhere''. The above is the most canonical definition, however, giving the unique unitary operator on {{math|''L''{{sup|2}}}} that is also an algebra homomorphism of {{math|''L''{{sup|1}}}} to {{math|''L''{{sup|∞}}}}.<ref>{{cite book |title=Harmonic analysis in phase space |first=Gerald |last=Folland |publisher=Princeton University Press |year=1989 |page=5 |author-link=Gerald Folland}}</ref>

The [[Heisenberg uncertainty principle]] also contains the number {{pi}}.  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,

<math display=block>
\left(\int_{-\infty}^\infty x^2|f(x)|^2\,dx\right)
\left(\int_{-\infty}^\infty \xi^2|\hat{f}(\xi)|^2\,d\xi\right)
 \ge
\left(\frac{1}{4\pi}\int_{-\infty}^\infty |f(x)|^2\,dx\right)^2.
</math>

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a [[quantum mechanical]] system, is [[#Describing physical phenomena|discussed below]].  The appearance of {{pi}} 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]].<ref name=howe>{{cite journal |first=Roger |last=Howe |title=On the role of the Heisenberg group in harmonic analysis |journal=[[Bulletin of the American Mathematical Society]] |volume=3 |pages=821–844 |number=2 |year=1980 |doi=10.1090/S0273-0979-1980-14825-9 |mr=578375 |doi-access=free}}</ref>