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Heisenberg Form of Uncertainty Relations

Received: 13 December 2015     Accepted: 21 December 2015     Published: 25 December 2015
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Abstract

This paper, deals with the uncertainty relation for photons. In [Phys.Rev.Let.108, 140401 (2012)], and [1] the uncertainty relation was obtained as a sharp inequality by using the energy distribution on space. The relation we obtain here is an alternative to the one given in [Phys.Rev.Let.108, 140401 (2012)] by the use of the position of the center of the energy operator. The fact that the components of the center is non commutative affected the right hand side of the Heisenberg inequality. But this resolved by the increase of the photon energy. Furthermore we study the uncertainty of Heisenberg with respect to angular momentum and Foureir. We end the paper by giving some examples.

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)
DOI 10.11648/j.ajam.20150306.22
Page(s) 321-326
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), 2015. Published by Science Publishing Group

Keywords

Uncertainty Relation for Photons, Quantum Mechanics of Photons, Foureir Theory

References
[1] Iwo Bialynicki-Birula∗. Zofia Bialynicka-Birula Heisenberg uncertainty relations for photons (2012), Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland.
[2] I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975), Chap. 9; I. Bialynicki-Birula and Z. Bialynicka-Birula, Phys. Rev. D35, 2383 (1987); I. Bialynicki-Birula and Z. Bialynicka-Birula, J. Opt. 13, 064014 (2011).
[3] U. M. Titulauer and R. J. Glauber, Phys. Rev. 145, 1041 (1966); Brian J. Smith and M. G. Raymer, New J. Phys.9, 414 (2007).
[4] R. J. Glauber, Phys. Rev. 130, 2529 (1963); L. Man-del and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, U.K., 1995).
[5] V. Garc´es-Ch´avez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, Phys. Rev. Lett. 91, 093602 (2003).
[6] McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto, 0-07-145546-9. McGraw-Hill eBooks. Quantum mechanics Demystified.Chap.4,5.
[7] Bialynicki-Birula I and Bialynicka-Birula Z 2006 Beams of electromagnetic radiation carrying angular momentum: the Riemann–Silberstein vector and the classical-quantum correspondence Opt.
[8] Bialynicki-Birula, photon wave function Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 1996), Vol.36; see also ArXiv: quant-ph/0508202.
[9] Bialynicki-Birula I and Bialynicka-Birula Z 1975 Quantum Electrodynamics (Oxford: Pergamon)
[10] “Generalized theory of interference, and its applications,” S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247-262 (1956). See also reference [42] below for the formulation of Pancharatnam’s phase in quantum theoretical language.
[11] “The Berry phase as an appropriate correspondence limit of the Aharonov-Anandan phase in a simple model,” J. Christian and A. Shimony, in Quantum Coherence, edited by J. Anandan (World Scientific, Singapore, 1990) pp 121-135.
[12] E.P. Wigner in Group Theory and its applications to the Quantum Mechanics of Atomic Spectra, Academic Press, NY(1959); A. Vaglica and G. Vetri, Optics Communications, 51,4(1984)239.
[13] J. Schwinger in Quantum theory of Angular Momentum, ed. L. Beidenharn and H. van Dam, Academic Press, NY.(1965)22 K.A. Milton, Rep. Prog. Phys.69.1637 (2006).
[14] K.A. Milton, Rep. Prog. Phys.69.1637 (2006).
[15] McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto, 0-07-145546-9. McGraw-Hill eBooks. Quantum mechanics Demystified. Chap.6-10.
[16] V.B. Berestetskii, E.M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynamics, 2nd ed. (Pergamon Press Ltd., NY, 1982).
[17] Birulaand Z. Bialynicka-Birula, J.Opt.13, 064014(2011).
[18] U.M. Titulauerand R.J. Glauber, Phys.Rev.145, 1041(1966); Brian J. Smithand M.G. Raymer, New J. Phys.9, 414(2007).
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    Mohammed Yousif, Mohammed Ali Basheir, Emadaldeen Abdalrahim. (2015). Heisenberg Form of Uncertainty Relations. American Journal of Applied Mathematics, 3(6), 321-326. https://doi.org/10.11648/j.ajam.20150306.22

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    ACS Style

    Mohammed Yousif; Mohammed Ali Basheir; Emadaldeen Abdalrahim. Heisenberg Form of Uncertainty Relations. Am. J. Appl. Math. 2015, 3(6), 321-326. doi: 10.11648/j.ajam.20150306.22

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    AMA Style

    Mohammed Yousif, Mohammed Ali Basheir, Emadaldeen Abdalrahim. Heisenberg Form of Uncertainty Relations. Am J Appl Math. 2015;3(6):321-326. doi: 10.11648/j.ajam.20150306.22

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  • @article{10.11648/j.ajam.20150306.22,
      author = {Mohammed Yousif and Mohammed Ali Basheir and Emadaldeen Abdalrahim},
      title = {Heisenberg Form of Uncertainty Relations},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {6},
      pages = {321-326},
      doi = {10.11648/j.ajam.20150306.22},
      url = {https://doi.org/10.11648/j.ajam.20150306.22},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.22},
      abstract = {This paper, deals with the uncertainty relation for photons. In [Phys.Rev.Let.108, 140401 (2012)], and [1] the uncertainty relation was obtained as a sharp inequality by using the energy distribution on space. The relation we obtain here is an alternative to the one given in [Phys.Rev.Let.108, 140401 (2012)] by the use of the position of the center of the energy operator. The fact that the components of the center is non commutative affected the right hand side of the Heisenberg inequality. But this resolved by the increase of the photon energy. Furthermore we study the uncertainty of Heisenberg with respect to angular momentum and Foureir. We end the paper by giving some examples.},
     year = {2015}
    }
    

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    T1  - Heisenberg Form of Uncertainty Relations
    AU  - Mohammed Yousif
    AU  - Mohammed Ali Basheir
    AU  - Emadaldeen Abdalrahim
    Y1  - 2015/12/25
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    DO  - 10.11648/j.ajam.20150306.22
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 326
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20150306.22
    AB  - This paper, deals with the uncertainty relation for photons. In [Phys.Rev.Let.108, 140401 (2012)], and [1] the uncertainty relation was obtained as a sharp inequality by using the energy distribution on space. The relation we obtain here is an alternative to the one given in [Phys.Rev.Let.108, 140401 (2012)] by the use of the position of the center of the energy operator. The fact that the components of the center is non commutative affected the right hand side of the Heisenberg inequality. But this resolved by the increase of the photon energy. Furthermore we study the uncertainty of Heisenberg with respect to angular momentum and Foureir. We end the paper by giving some examples.
    VL  - 3
    IS  - 6
    ER  - 

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Author Information
  • Sudan University of Science and Technology (SUST), Math Department, College of Science, Khartoum, Sudan

  • Math Department, College of Science, ALNeelein University, Khartoum, Sudan

  • Sudan University of Science and Technology (SUST), Math Department, College of Science, Khartoum, Sudan

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