Heisenberg Form of Uncertainty Relations

Mohammed Yousif; Mohammed Ali Basheir; Emadaldeen Abdalrahim

doi:doi:10.11648/j.ajam.20150306.22

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Heisenberg Form of Uncertainty Relations

Mohammed Yousif, Mohammed Ali Basheir, Emadaldeen Abdalrahim

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)

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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APA Style

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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TY  - JOUR
T1  - Heisenberg Form of Uncertainty Relations
AU  - Mohammed Yousif
AU  - Mohammed Ali Basheir
AU  - Emadaldeen Abdalrahim
Y1  - 2015/12/25
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.22
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
SP  - 321
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

Mohammed Yousif

Sudan University of Science and Technology (SUST), Math Department, College of Science, Khartoum, Sudan
Mohammed Ali Basheir

Math Department, College of Science, ALNeelein University, Khartoum, Sudan
Emadaldeen Abdalrahim

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

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APA Style

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Heisenberg Form of Uncertainty Relations
AU  - Mohammed Yousif
AU  - Mohammed Ali Basheir
AU  - Emadaldeen Abdalrahim
Y1  - 2015/12/25
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.22
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
SP  - 321
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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