Download THE UNCERTAINTY PRINCIPLE The uncertainty principle states

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Transcript
THE UNCERTAINTY PRINCIPLE
MICHAEL G COWLING
The uncertainty principle states that a function f on Rn and its Fourier transform fˆ can
not both be too highly localized. In the Schrödinger formulation of quantum mechanics,
this becomes the statement that both the position and the momentum of a quantum
particle cannot be prescribed too accurately, and if we think of the Fourier transform as
giving a spectral version of the function, then the principle states that a function and its
spectrum cannot both be too highly localized. One variation of the principle, sometimes
ascribed to Heisenberg, Pauli and Weyl, states that
Z
2
Z
Z
2
2
2
|f (x)| dx ≤ C |x − a| |f (x)| dx
|ξ − α|2 |fˆ(ξ)|2 dξ
R
R
R
and another, due to Hardy, states that if
|f (x)| ≤ Ce−a|x|
2
and
|fˆ(ξ)| ≤ Ce−α|ξ|
2
where a, α > 0, and if aα is large enough, then f has to be zero.
These and similar results have been generalised into different areas, such as spectral
analysis of functions on manifolds and the analysis of operators. This talk reviews the
original theorems, and discusses recent developments of the theory.
1