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On the Justification and Validity of the Kennard Inequality

Received: 5 November 2020     Accepted: 23 November 2020     Published: 4 December 2020
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Abstract

In 1927, Earle Hesse Kennard derived an inequality describing Heisenberg’s uncertainty principle. Since then, we have traditionally been using the standard deviation as the measure of uncertainty in quantum mechanics. But Jan Hilgevoord asserts that the standard deviation is neither a natural nor a generally adequate measure of quantum uncertainty. Specifically, he asserts that the standard deviations are inadequate to use as the quantum uncertainties in the single- and double-slit diffraction experiments. He even tells that from these examples it will become clear that the standard deviation is the wrong concept to express the uncertainty principle generally and that the Kennard relation has little to do with the uncertainty principle. We will investigate what are adequate as the measures of quantum uncertainty. And, beyond that, we will investigate the effects of multiplying the two uncertainties; namely, characteristics which is hiding in deep interior of the Kennard inequality. Through investigations we’ll come to naturally realize that his assertions were wrong. All of our discussions will help raise understanding of the Heisenberg uncertainty principle. Our discussions will afford us an opportunity to think about the essence of the Fourier transform. The aim of this paper is to draw conclusions about whether the Kennard inequality is justified or not.

Published in American Journal of Modern Physics (Volume 9, Issue 5)
DOI 10.11648/j.ajmp.20200905.12
Page(s) 73-76
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Kennard Inequality, Quantum Uncertainty, Uncertainty Relation, Uncertainty Principle, Absolute Deviation, Standard Deviation

References
[1] W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Physik, vol. 43, pp. 172–198, March 1927.
[2] E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Z. Physik, vol. 44, pp. 326–352, April 1927.
[3] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
[4] E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
[5] J. J. Sakurai, Modern Quantum Mechanics, Rev. ed., Addison-Wesley, 1994.
[6] M. A. de Gosson, The Principles of Newtonian and Quantum Mechanics, 2nd ed., World Scientific, 2017.
[7] A. Kumar, Fundamentals of Quantum Mechanics, Cambridge University Press, 2018.
[8] J. Baggott, The Quantum Cookbook, Oxford University Press, 1st ed., 2020.
[9] P. L. Bowers, Lectures on Quantum Mechanics, Cambridge University Press, 2020.
[10] S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 3 ed., Springer, 2020.
[11] J. Izaac and J. Wang, Computational Quantum Mechanics, Springer, 2018.
[12] J. B. M. Uffink, J. Hilgevoord, “Uncertainty principle and uncertainty relations,” J. Found. Phys, vol. 15, pp. 925–944, September 1985.
[13] J. Hilgevoord, “The standard deviation is not an adequate measure of quantum uncertainty,” Am. J. Phys, vol. 70, pp. 983, October 2002.
[14] B. L. Agarwal, Basic Statistics, 4th ed., New Age International Publishers, 2006, pp. 61–81.
[15] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Elsevier, 2013.
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    Haengjin Choe. (2020). On the Justification and Validity of the Kennard Inequality. American Journal of Modern Physics, 9(5), 73-76. https://doi.org/10.11648/j.ajmp.20200905.12

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    ACS Style

    Haengjin Choe. On the Justification and Validity of the Kennard Inequality. Am. J. Mod. Phys. 2020, 9(5), 73-76. doi: 10.11648/j.ajmp.20200905.12

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    AMA Style

    Haengjin Choe. On the Justification and Validity of the Kennard Inequality. Am J Mod Phys. 2020;9(5):73-76. doi: 10.11648/j.ajmp.20200905.12

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  • @article{10.11648/j.ajmp.20200905.12,
      author = {Haengjin Choe},
      title = {On the Justification and Validity of the Kennard Inequality},
      journal = {American Journal of Modern Physics},
      volume = {9},
      number = {5},
      pages = {73-76},
      doi = {10.11648/j.ajmp.20200905.12},
      url = {https://doi.org/10.11648/j.ajmp.20200905.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20200905.12},
      abstract = {In 1927, Earle Hesse Kennard derived an inequality describing Heisenberg’s uncertainty principle. Since then, we have traditionally been using the standard deviation as the measure of uncertainty in quantum mechanics. But Jan Hilgevoord asserts that the standard deviation is neither a natural nor a generally adequate measure of quantum uncertainty. Specifically, he asserts that the standard deviations are inadequate to use as the quantum uncertainties in the single- and double-slit diffraction experiments. He even tells that from these examples it will become clear that the standard deviation is the wrong concept to express the uncertainty principle generally and that the Kennard relation has little to do with the uncertainty principle. We will investigate what are adequate as the measures of quantum uncertainty. And, beyond that, we will investigate the effects of multiplying the two uncertainties; namely, characteristics which is hiding in deep interior of the Kennard inequality. Through investigations we’ll come to naturally realize that his assertions were wrong. All of our discussions will help raise understanding of the Heisenberg uncertainty principle. Our discussions will afford us an opportunity to think about the essence of the Fourier transform. The aim of this paper is to draw conclusions about whether the Kennard inequality is justified or not.},
     year = {2020}
    }
    

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Author Information
  • Department of Physics, College of Natural Sciences, Chonnam National University, Gwangju, South Korea

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