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Journal of Physics A Mathematical and General

On a proposed new test of Heisenberg's principle

M C Robinson

Published Mar 1, 1980• 6 citations
Publisher

Abstract

The design of a technically feasible experiment is discussed that would distinguish between Heisenberg's uncertainty principle and de Broglie's relation, ( Delta x) i ( Delta p) f >or=h(cross)/2 where ( Delta x) i is the initial localisation of the particle and ( Delta p) f is the final scatter in the momentum of the free particle.

DOI
10.1088/0305-4470/13/3/020

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Reply to comments on 'On a proposed new test of Heisenberg's principle'

M C Robinson

It is shown that the claims of Home and Sengupta (1981) and of Singh (1981) to have discovered fallacies in the author's analysis of a proposed, technically feasible test of Heisenberg's uncertainty principle are based on misunderstandings.

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The design of a technically feasible experiment is discussed that would distinguish between Heisenberg's uncertainty principle and de Broglie's relation, ( Delta x) i ( Delta p) f >or=h(cross)/2 where ( Delta x) i is the initial localisation of the particle and ( Delta p) f is the final scatter in the momentum of the free particle.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
If you asked members of the public to name a concept or idea from modern physics, the most popular answer would probably be E = mc2. However, the runner-up could well be Heisenberg's uncertainty principle. This principle states that the position and linear momentum of a particle cannot be known simultaneously with arbitrary precision. More precisely, it sets a lower bound for the product of the uncertainties of these quantities: ΔxΔp≥/2, where Δx is the uncertainty in the position, Δp is the uncertainty in the momentum and is Planck's constant divided by 2π.
Based on the hypothesis that the position-representation of a physical state <x|α\alphaα> is the Fourier transform of its momentum-representation <P|α\alphaα> and that the time-representation <t|α\alphaα> is the inverse Fourier transform of its energy-representation <E|α\alphaα>, we are able to find the Planck relation E = hv, the de Broglie relation P = hλ\frac{h}{\lambda}λh​. Afterward, utilizing the Dirac delta function δ\deltaδ(x) and the property FTxf(x) = -iℏ\hbarℏ∂\partial∂, pFTxf(x) = -iℏ\hbarℏ∂\partial∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^\hat{p}, \hat{x}p^​,x^] = -iℏ\hbarℏI^\hat{I}I^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.