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Physical Review

Note on Heisenberg's Indetermination Principle

Earle H. Kennard

Published Mar 1, 1928• 10 citations
Publisher

Abstract

The indetermination principle of Heisenberg is discussed and applied to cases where position and velocity of an electron are observed by allowing it to pass through shutters opened momentarily at known times. The principle requires that such a shutter shall change the speed of the electron in a manner that can only be predicted statistically, in the same way as the frequency of a light quantum passing through such a shutter must usually undergo alteration to correspond with the Fourier resolution of the finite train of waves that gets through while the shutter is open. This effect is small and hard to detect experimentally.

DOI
10.1103/physrev.31.344

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The indetermination principle of Heisenberg is discussed and applied to cases where position and velocity of an electron are observed by allowing it to pass through shutters opened momentarily at known times. The principle requires that such a shutter shall change the speed of the electron in a manner that can only be predicted statistically, in the same way as the frequency of a light quantum passing through such a shutter must usually undergo alteration to correspond with the Fourier resolution of the finite train of waves that gets through while the shutter is open. This effect is small and hard to detect experimentally.
Until Heisenberg proposed the uncertainty principle in 1927, it was an open question whether models for the structure of matter reflected actual reality or were useful fictions for operational purposes. There wasn’t any sense in either case that there might be parameters it was impossible to determine. The uncertainty principle says that it is impossible to determine simultaneously the location and momentum of an electron. While it applies to a specific situation, ‘uncertainty’ is a good description for the entire approach of quantum theory. Schrödinger’s equations show that electrons behave simultaneously as waves and particles. The double-slit experiment, when a beam of light or electrons passes through two parallel slits, shows that the act of observation influences the behavior of the wave–particle duality. The ability of a pair of ‘entangled’ electrons (or other particles) to influence each other instantaneously appears to exceed the speed limit of light. Paradoxes of quantum behavior may even extend to biological situations. Intuition or common sense are no guide to quantum behavior.
The essential indefiniteness of quantum mechanics {Heisenberg) The origin of quantum mechanics was an attempt to break away from the usual kinematical and mechanical concepts and in their place to substitute relations between concrete numbers given by experiment. The position of an electron The position of an electron relative to a given system of axes must be measurable by experiment if the term is to have a meaning in quantum mechanics. We illuminate the electron and observe it through a microscope; the accuracy with which the position can be observed is given by the wave-length of the light used. If we use a y ray microscope, we should get the most exact result physically possible. But these y rays produce a Compton effect on the electron, so that at the instant when the position is determined the momentum of the electron is suddenly changed. This change of momentum is all the greater the shorter the wave-length of the light used, i.e. the more accurate the determination of position is. Thus at the moment when the position of the electron is known, its momentum can only be known to an order equal to this discontinuous change. Hence the more exactly a coordinate q is determined, the less exactly can its momentum p be found; and conversely.
Light quanta are treated by wave mechanics by analogy with electrons. It is shown that their wave equation is the ordinary optical wave equation. Heisenberg's principle of indeterminateness becomes a description of diffraction. In problems in which localization of quanta is found experimentally, wave packets are to be set up; these are applied to the experiments of Bothe and Geiger, and of Compton. The paths of quanta coincide with the rays of geometrical optics, with a deviation of the error in geometrical optics; by the principle of indeterminateness, more accurate laws for the paths are neither necessary nor possible.
The Heisenberg equation may not be as well-known as Einstein’s e=mc^2. But the Heisenberg equation’s implications that there is an irreducible uncertainty with regard to the position of an object are no less intriguing than the implications of Einstein’s relativity equation that time slows to a stop when the relative velocity of an object approaches the speed of light. Using elementary mathematics and science, this paper begins by examining the physics of motion of an everyday object such as a tennis ball. The paper shows that the same physics that applies to everyday objects in fact also applies when it comes to subatomic particles, such as an electron in motion. And by taking gradual incremental steps in developing the mathematics of the underlying physics, the paper leads to the conclusion that the universe is inherently uncertain, that it is impossible to know the exact position of any object.
The Heisenberg equation may not be as well-known as Einstein’s e=mc^2. But the Heisenberg equation’s implications that there is an irreducible uncertainty with regard to the position of an object are no less intriguing than the implications of Einstein’s relativity equation that time slows to a stop when the relative velocity of an object approaches the speed of light. Using elementary mathematics and science, this paper begins by examining the physics of motion of an everyday object such as a tennis ball. The paper shows that the same physics that applies to everyday objects in fact also applies when it comes to subatomic particles, such as an electron in motion. And by taking gradual incremental steps in developing the mathematics of the underlying physics, the paper leads to the conclusion that the universe is inherently uncertain, that it is impossible to know the exact position of any object.
According to Heisenberg’s uncertainty principle [1], both, position and momentum of a particle, cannot be known with ultimate accuracy; the product: Δ x . Δ p > h . Whereas in the case of Laser-cooled atoms, we know that velocity of the particle is close to zero; and its position x is so perfectly known, that an atom is said to be held in Laser forceps [2]. These experiments show that both position and momentum of atoms are accurately knowable simultaneously! Similarly, when momentum of an atom is zero, the wavelength of its de Broglie wave is very very long; and according to quantum-mechanical interpretation, the atom is likely to be detected anywhere within its de Broglie wavelength, with higher probability at the peaks of the wave. Whereas the experiments with Laser-cooled atoms show that the atom is confined precisely within the Laser tweezers. Based on this observation a new experiment is proposed here in which, instead of quantum mechanical waves, which have wavelength h / m v , corresponding to phase velocity of the wave, we can carefully observe the waves corresponding to group-velocity, of the wavelength h v / m c^2. And we can expect that a particle may deterministically tunnel from one peak of the group-wave to next peak of the group wave; and can be detected at deterministically predictable points.