From: Charles Francis Subject: Re: Uncertainty Principle and Fourier Analysis Date: Wed, 5 Jan 2000 07:23:52 +0000 Newsgroups: sci.physics,sci.math Summary: [missing] In article , Paul M writes >Hello. From most of the quantum mechanics books that I've read, it >is stated that the Heisenberg Uncertainty Principle is really nothing >more than a simple corollary from a well known result in Fourier >analysis that (Dx)(Dk) >= 1, where D stands for delta. (The >connection is made by the principle of wave-particle duality.) Now, >every book in mathematical physics and Fourier analysis that I've >consulted contain no proof of this more general fact. Would anyone >happen to know of a refernce with a proof of this statement in Fourier >analysis? (I'm aware of a purely algebraic proof of the generalized >uncertainty realtions for canonically conjugate observables, but >that's not what I'm looking for.) > >Thank you very much. > >- Paul I had not heard it suggested that the inequality is a simple corollary of a well known result. It was first proved from Fourier analysis by E. H. Kennard, Zeitschrift für Physik, 44, 326 (1927) though credit is usually given for a more formal and general proof, to H.P. Robertson, Phys Rev. 34, 163 (1929) -- Regards Charles Francis charles@clef.demon.co.uk ============================================================================== From: Valery P Dmitriyev Subject: Re: Uncertainty Principle and Fourier Analysis Date: Wed, 05 Jan 2000 17:52:24 GMT Newsgroups: sci.physics,sci.math In article , Charles Francis wrote: > In article , Paul M > writes > >Hello. From most of the quantum mechanics books that I've read, it > >is stated that the Heisenberg Uncertainty Principle is really nothing > >more than a simple corollary from a well known result in Fourier > >analysis that (Dx)(Dk) >= 1, where D stands for delta. (The > >connection is made by the principle of wave-particle duality.) Now, > >every book in mathematical physics and Fourier analysis that I've > >consulted contain no proof of this more general fact. Would anyone > >happen to know of a refernce with a proof of this statement in Fourier > >analysis? (I'm aware of a purely algebraic proof of the generalized > >uncertainty realtions for canonically conjugate observables, but > >that's not what I'm looking for.) > > > >Thank you very much. > > > >- Paul > This is indeed so. The so called uncertainty principle follows simply from the Fourier analysis of the wave function. See e.g. the commentaries to Enrico Fermi "Notes on quantum mechanics" where the proof is given after E.Persico text-book . The math that you seek for can be found not in the books on Fourier analysis but in the text-books on the probability theory. See the proof of the central limit theorem via the characteristic functions. The point is in the following. Take the Fourier transform of the Gauss distribution. The result will be again a Gauss distribution function but now for the parameter of the transform. Its "variance" appears to be inversely proportional to the variance of the original distribution. Now multiply the both and you'll get a constant. That is all content of the uncertainty principle: the product of the two variances is constant. One must agree that this is rather a formal construction. The more reasonable "derivation" of the uncertainty relation can be found in stochastic mechanics (i.e. in quantum mechanics represented in the form of the theory of a random process) . In this approach it is stated as the covariance of the two random variables -- the coordinate and the drift (i.e. the diffusion velocity) taken as a function of the random coordinate [1]. The reference is given to the Russian book. It is inaccessible (if only exists in Spanish translation). However, all relevant mathematics can be found at http://publish.aps.org/eprint/gateway/eplist/aps1999apr24_003 http://xxx.lanl.gov/abs/physics/9904034 or equally in [2], section 5. [1] V.P.Dmitriyev, Stochastic mechanics, Vyssaja Scola, Moscow, 1990. [2] V.P.Dmitriyev, Turbulent advection of a fluid discontinuity and Schroedinger mechanics, Galilean Electrodynamics, vol.10, No 5, pp.95-99 (1999). Sent via Deja.com http://www.deja.com/ Before you buy.