Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect

Gilbert A. Ibitola; Olanrewaju Ajanaku

doi:doi:10.11648/j.wjap.20160102.15

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Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect

Gilbert A. Ibitola, Olanrewaju Ajanaku

Published in World Journal of Applied Physics (Volume 1, Issue 2)

Received: 19 October 2016 Accepted: 4 November 2016 Published: 20 December 2016

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Abstract

In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.

Published in	World Journal of Applied Physics (Volume 1, Issue 2)
DOI	10.11648/j.wjap.20160102.15
Page(s)	59-66
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), 2024. Published by Science Publishing Group

Keywords

Schrodinger Wave Equation (SWE), Potential Step, Potential Barrier, Potential Well, Wave Function, Reflection Coefficient, Transmission Probability and Tunneling Effect

References

[1]	Maduemezia, A. and Coker, A. J.: “Elements of Quantum Mechanics”, Beaver Publications, Ibadan, pp. 186-189 (2003).
[2]	Jeffries, H., Proc., London Math. Soc., 23, 428 (1983).
[3]	Wentzel, G., Zeitschrift fur Physik, 38, 508 (1986).
[4]	Kramers, H. A., Zeitschrift fur Physik 39, 828 (1990).
[5]	Brillouin, L., Journal de Physique 7, 353 (1998).
[6]	Leighton, R. B., Principles of Modern Physics, McGraw-Hill, 2004.
[7]	Anderson, E. E., Modern Physics and Quantum Mechanics, W.B. Saunders, 1991.
[8]	Griffiths, D. J., Introduction to Quantum mechanics – Second Edition, Person Prentice Hall, Inc., 2005.
[9]	Scherrer, R., Quantum Mechanics: An Accessible Introduction, Addison Wesley, 2006.
[10]	Winter, R, G., Quantum Physics, Faculty Publishing, Inc., Davis, California, 1996.
[11]	Zettili, N., Quantum Mechanics – Concepts and Applications, John Wiley & Sons, Inc., 2001.
[12]	Halperin, B. I., 1992, Phys. Rev. B 25, 2185.
[13]	Halperin, B. I., 1993, Helv. Phys. Acta 56, 75.
[14]	Halperin, B. I., 1994, Phys. Rev. Lett. 52, 1583.
[15]	Halperin, B. I., T. C. Lubensky, and S. K. Ma, 1994, Phys. Rev. Lett. 32, 292.
[16]	Hiromichi Nakazato, Mikio Namiki, and Saverio Pascazio (2012): “Temporal Behaviour Of Quantum Mechanical Systems”, Waseda Univ. Publ.
[17]	Gupta Kumar (2009): “Principles of Physical Electronics”, S. Chand & Co., pp. 123-137.
[18]	Sophya Garashchuk, Bing Gu, and James Mazzuca (2014): “Calculation of the quantum-mechanical tunneling in bound potentials”, Hindawi Publishing Co., J. Theor. Chem. Phys., Vol. 2014, Art ID 240491, 1-11.
[19]	Http://www.quantum-physics.com/’quantum-mechanical -potentials’/ (September, 2015).

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APA Style

Gilbert A. Ibitola, Olanrewaju Ajanaku. (2016). Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World Journal of Applied Physics, 1(2), 59-66. https://doi.org/10.11648/j.wjap.20160102.15

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ACS Style

Gilbert A. Ibitola; Olanrewaju Ajanaku. Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World J. Appl. Phys. 2016, 1(2), 59-66. doi: 10.11648/j.wjap.20160102.15

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AMA Style

Gilbert A. Ibitola, Olanrewaju Ajanaku. Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World J Appl Phys. 2016;1(2):59-66. doi: 10.11648/j.wjap.20160102.15

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@article{10.11648/j.wjap.20160102.15,
  author = {Gilbert A. Ibitola and Olanrewaju Ajanaku},
  title = {Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect},
  journal = {World Journal of Applied Physics},
  volume = {1},
  number = {2},
  pages = {59-66},
  doi = {10.11648/j.wjap.20160102.15},
  url = {https://doi.org/10.11648/j.wjap.20160102.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20160102.15},
  abstract = {In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.},
 year = {2016}
}

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TY  - JOUR
T1  - Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect
AU  - Gilbert A. Ibitola
AU  - Olanrewaju Ajanaku
Y1  - 2016/12/20
PY  - 2016
N1  - https://doi.org/10.11648/j.wjap.20160102.15
DO  - 10.11648/j.wjap.20160102.15
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
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EP  - 66
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20160102.15
AB  - In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
VL  - 1
IS  - 2
ER  -

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Author Information

Gilbert A. Ibitola

Department of Physical Sciences, Ondo State University of Science and Technology, Okitipupa, Nigeria
Olanrewaju Ajanaku

Department of Physical Sciences, Ondo State University of Science and Technology, Okitipupa, Nigeria

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APA Style

Gilbert A. Ibitola, Olanrewaju Ajanaku. (2016). Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World Journal of Applied Physics, 1(2), 59-66. https://doi.org/10.11648/j.wjap.20160102.15

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ACS Style

Gilbert A. Ibitola; Olanrewaju Ajanaku. Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World J. Appl. Phys. 2016, 1(2), 59-66. doi: 10.11648/j.wjap.20160102.15

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AMA Style

Gilbert A. Ibitola, Olanrewaju Ajanaku. Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect. World J Appl Phys. 2016;1(2):59-66. doi: 10.11648/j.wjap.20160102.15

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@article{10.11648/j.wjap.20160102.15,
  author = {Gilbert A. Ibitola and Olanrewaju Ajanaku},
  title = {Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect},
  journal = {World Journal of Applied Physics},
  volume = {1},
  number = {2},
  pages = {59-66},
  doi = {10.11648/j.wjap.20160102.15},
  url = {https://doi.org/10.11648/j.wjap.20160102.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20160102.15},
  abstract = {In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect
AU  - Gilbert A. Ibitola
AU  - Olanrewaju Ajanaku
Y1  - 2016/12/20
PY  - 2016
N1  - https://doi.org/10.11648/j.wjap.20160102.15
DO  - 10.11648/j.wjap.20160102.15
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 59
EP  - 66
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20160102.15
AB  - In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
VL  - 1
IS  - 2
ER  -

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