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arXiv:0812.4775

Single-slit diffraction and the Heisenberg principle for position and momentum

Winfrid Görlich, Ingo Hoffmann, Thomas Schürmann

Published Dec 27, 2008• 0 citations
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Abstract

Monochromatic light is prepared by a single slit of spatial width Δx. The diffraction pattern is considered to obtain the corresponding probability density of the momentum. Let P be the probability weight to measure a momentum in a finite window of width Δp. For small Δp the asymptotic law P ~ ΔxΔp/h is verified, while h is Planck's constant of action.

DOI
10.48550/arxiv.0812.4775

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Monochromatic light is prepared by a single slit of spatial width Δx. The diffraction pattern is considered to obtain the corresponding probability density of the momentum. Let P be the probability weight to measure a momentum in a finite window of width Δp. For small Δp the asymptotic law P ~ ΔxΔp/h is verified, while h is Planck's constant of action.
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We show that the quantum mechanical interpretation of the diffraction of light on a slit, when a wave function is assigned to a photon, can be used for a direct experimental study of Heisenberg’s position-momentum and equivalent positionwave vector uncertainty relation for the photon. Results of an experimental test of the position-wave vector uncertainty relation, where the wavelength is used as the input parameter, are given and they very well confirm our approach. The same experimental results can also be used for a test of the position-momentum uncertainty relation when the momentum p0 of a photon is known as the input parameter. We show that a measurement of p0, independent of the knowledge of the value of the Planck’s constant, is possible. Using that value of p0, a test of the position-momentum uncertainty relation could be regarded as a method for a direct measurement of the Planck’s constant. This is discussed, since the diffraction pattern is also well described by classical electrodynamics in the considered experimental conditions. This approach for testing the Heisenberg’s uncertainty relations is very simple and consequently suitable as a quantitative exercise in undergraduate experimental courses, as well as a visual and attractive demonstration of the Heisenberg’s uncertainty principle in courses of quantum mechanics.
A thought experiment, in which a light beam is sent through slits of different sizes thereby producing a diffraction pattern, is used to illustrate the uncertainty principle. Attempts to reduce the uncertainty in the measurement of the position of the light beam ( ) succeed, but at the expense of increasing the uncertainty in the measurement of the momentum ( ), so that the product of the two magnitudes, x, is still larger than h, Planck's constant. This experiment introduces the idea that one can know the value of a physical observable with as high a precision as necessary, but the act of measuring this observable has an effect on how much we can know about another observable. This thought experiment can be physically performed using a diode laser and a series of slits. A derivation is presented to relate the uncertainties in position and momentum to variables that can be measured with a ruler in a classroom or laboratory. Data from a typical experiment are also presented.
In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.
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We described a simple idea for experimental verification of the uncertainty principle for light waves. We used a single-slit diffraction of a laser beam for measuring the angular width of zero-order diffraction maximum and obtained the corresponding wave number uncertainty. We will assume that the uncertainty in position is the slit width. For the acquisition of the experimental data and their further analysis, we used a computer. Because of its simplicity this experiment is very suitable for demonstration, as well as for a quantitative exercise at universities and final year of high school studies.