Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation

Mash-Huda Rahman Shipra; Syed Badiuzzaman Faruque

doi:doi:10.11648/j.ijhep.20200702.11

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Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque

Published in International Journal of High Energy Physics (Volume 7, Issue 2)

Received: 7 July 2020 Accepted: 27 July 2020 Published: 10 August 2020

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Abstract

The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.

Published in	International Journal of High Energy Physics (Volume 7, Issue 2)
DOI	10.11648/j.ijhep.20200702.11
Page(s)	32-36
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

Modified Dirac Equation, Localized Wave Packet, Generalized Momentum, Logarithmic Coupling

References

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[6]	Bialynicki-Birula, I., & Bialynicka-Birula, Z. (2017). Relativistic electron wave packets carrying angular momentum. Physical review letters, 118 (11), 114801.
[7]	Park, S. T. (2012). Propagation of a relativistic electron wave packet in the Dirac equation. Physical Review A, 86 (6), 062105.
[8]	Strange, P. (2017). Angular momentum of a relativistic wave packet. Physical Review A, 96 (5), 052103.
[9]	Szabó, L. Z., Benedict, M. G., Czirják, A., & Földi, P. (2013). Relativistic electron transport through an oscillating barrier: Wave-packet generation and Fano-type resonances. Physical Review B, 88 (7), 075438.
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[11]	Demikhovskii, V. Y., Maksimova, G. M., & Frolova, E. V. (2008). Wave packet dynamics in a two-dimensional electron gas with spin orbit coupling: Splitting and zitterbewegung. Physical Review B, 78 (11), 115401.
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APA Style

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque. (2020). Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. International Journal of High Energy Physics, 7(2), 32-36. https://doi.org/10.11648/j.ijhep.20200702.11

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ACS Style

Mash-Huda Rahman Shipra; Syed Badiuzzaman Faruque. Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. Int. J. High Energy Phys. 2020, 7(2), 32-36. doi: 10.11648/j.ijhep.20200702.11

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AMA Style

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque. Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. Int J High Energy Phys. 2020;7(2):32-36. doi: 10.11648/j.ijhep.20200702.11

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@article{10.11648/j.ijhep.20200702.11,
  author = {Mash-Huda Rahman Shipra and Syed Badiuzzaman Faruque},
  title = {Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation},
  journal = {International Journal of High Energy Physics},
  volume = {7},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.ijhep.20200702.11},
  url = {https://doi.org/10.11648/j.ijhep.20200702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijhep.20200702.11},
  abstract = {The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.},
 year = {2020}
}

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TY  - JOUR
T1  - Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation
AU  - Mash-Huda Rahman Shipra
AU  - Syed Badiuzzaman Faruque
Y1  - 2020/08/10
PY  - 2020
N1  - https://doi.org/10.11648/j.ijhep.20200702.11
DO  - 10.11648/j.ijhep.20200702.11
T2  - International Journal of High Energy Physics
JF  - International Journal of High Energy Physics
JO  - International Journal of High Energy Physics
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2376-7448
UR  - https://doi.org/10.11648/j.ijhep.20200702.11
AB  - The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.
VL  - 7
IS  - 2
ER  -

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Author Information

Mash-Huda Rahman Shipra

Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh
Syed Badiuzzaman Faruque

Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

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APA Style

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque. (2020). Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. International Journal of High Energy Physics, 7(2), 32-36. https://doi.org/10.11648/j.ijhep.20200702.11

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ACS Style

Mash-Huda Rahman Shipra; Syed Badiuzzaman Faruque. Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. Int. J. High Energy Phys. 2020, 7(2), 32-36. doi: 10.11648/j.ijhep.20200702.11

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AMA Style

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque. Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation. Int J High Energy Phys. 2020;7(2):32-36. doi: 10.11648/j.ijhep.20200702.11

Copy | Download

@article{10.11648/j.ijhep.20200702.11,
  author = {Mash-Huda Rahman Shipra and Syed Badiuzzaman Faruque},
  title = {Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation},
  journal = {International Journal of High Energy Physics},
  volume = {7},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.ijhep.20200702.11},
  url = {https://doi.org/10.11648/j.ijhep.20200702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijhep.20200702.11},
  abstract = {The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation
AU  - Mash-Huda Rahman Shipra
AU  - Syed Badiuzzaman Faruque
Y1  - 2020/08/10
PY  - 2020
N1  - https://doi.org/10.11648/j.ijhep.20200702.11
DO  - 10.11648/j.ijhep.20200702.11
T2  - International Journal of High Energy Physics
JF  - International Journal of High Energy Physics
JO  - International Journal of High Energy Physics
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2376-7448
UR  - https://doi.org/10.11648/j.ijhep.20200702.11
AB  - The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.
VL  - 7
IS  - 2
ER  -

Copy | Download