Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

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

Document related concepts

Probability amplitude wikipedia, lookup

Coherent states wikipedia, lookup

Double-slit experiment wikipedia, lookup

Bohr–Einstein debates wikipedia, lookup

Quantum key distribution wikipedia, lookup

Wave function wikipedia, lookup

Copenhagen interpretation wikipedia, lookup

Wave–particle duality wikipedia, lookup

History of quantum field theory wikipedia, lookup

Canonical quantization wikipedia, lookup

Quantum state wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Path integral formulation wikipedia, lookup

Hidden variable theory wikipedia, lookup

Scalar field theory wikipedia, lookup

Renormalization group wikipedia, lookup

Quantum group wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Spectral density wikipedia, lookup

Many-worlds interpretation wikipedia, lookup

Orchestrated objective reduction wikipedia, lookup

Renormalization wikipedia, lookup

Bell's theorem wikipedia, lookup

Quantum teleportation wikipedia, lookup