* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download THE UNCERTAINTY PRINCIPLE The uncertainty principle states
Quantum group wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Quantum key distribution wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum teleportation wikipedia , lookup
Renormalization wikipedia , lookup
Double-slit experiment wikipedia , lookup
Matter wave wikipedia , lookup
Bell's theorem wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Probability amplitude wikipedia , lookup
Coherent states wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Wave function wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Scalar field theory wikipedia , lookup
Spectral density wikipedia , lookup
Wave–particle duality wikipedia , lookup
Particle in a box wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum state wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Renormalization group wikipedia , lookup
Path integral formulation wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Canonical quantization wikipedia , lookup
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