The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.
Published in | American Journal of Modern Physics (Volume 13, Issue 3) |
DOI | 10.11648/j.ajmp.20241303.11 |
Page(s) | 34-40 |
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 |
Analysis, Zitterbewegung, Free Particle, Dirac Equation, Heisenberg’s Representation, Relativistic Velocity, Fluctuation, Oscillatory Motion, Eigenvalues
[1] | B. Thaller (2010), The Dirac Equation. Springer Verlag ISBN 978-3-642-08134-7. |
[2] | J. P Antoine (2004) Relativistic Quantum Mechanics. J. Phy A 37 (4) |
[3] | Y. V. Nazarov and J. Danon (2013) Advance Quantum Mechanics; Practical Guide. Cambridge University Press. |
[4] | F. Dyson (2011) Advance Quantum Mechanics Second Edition. World Scientific ISBN 978-981-4383-40-0. |
[5] | A. Niehaus (2016) Foundation of Physics, 16, 3-13 |
[6] | V. F. Lazutkin (1993) Theory and Semi-classical Approximation to Eigenfunction. Ergebnisseder Mathematic. Vol. 24, Springer-Verlag. |
[7] | L. P Horwitz (2007). Quantum interference in time. Foundations of Physics 37 pp 734-746. |
[8] | D. Hestenes (1979) Uncertainty in interpretation of Quantum Mechanics. American Journal of Physics. 47. 399-418. |
[9] | R. Winkler, U. Zu’’licke and J. Bolte, (2007). Oscillatory multiband dynamics of free particles: The ubiquity of zitterbewegung effect. Phy. Review B. 75, 205314. |
[10] | J. S Briggs and J. M. Rost (2001), On the Derivation of the Time- Dependent Equation of Schrodinger. Foundation of Physics 31 pp 693-712. |
[11] | K. Haung (1956), On The Zitterbewegung of the Electron. Am. J. Phy. 20. pp. 479-484. |
[12] | A. O. Barut and A. J Backen (1981). Zitterbewegung and Internal Geometry of the electron. Phy. RevD 23.2454. |
[13] | B. G. Sidharth (2009). Revisiting Zitterbewegung. Int. J. nTheor. Phy. 48 pp 497-506. |
[14] | T. M. Rusin and W. Zawadski (2009) Theory of electron Zitterbewegung in graphen probe by Femto-second laser pulse. Phys Rev. B 80, 045416. |
[15] | J. Bolte and S. Keppler (1999) A Semi-Classical Approach to the Dirac Equation. Ann. Phys.(NY) 274, 125-162. |
[16] | A. Niehaus (2017) Journal of modern Physics. Scientific Research publishing. |
[17] | W. Greiner, (2000). Relativistic Quantum Mechanics, Third Edition. Spinger Verlag. |
[18] | J. S. Briggs and J. M Rost, (2000). Time dependence in Quantum Mechanics. Eur. Phy. J. D 10. pp. 311-318. |
[19] | J. G Muga (2009). Time in Quantum Mechanics. Vol. 2. Springer-Verlag. |
[20] | M. Bauer (1983). A Time Operator in Quantum Mechanic. Ann. Phy. (N. Y) 150 pp 1-21. |
[21] | W. Pauli (1980) The general Principles of Quantum Mechanics. Springer- Verlag Berlin. |
[22] | G. B Folland (1989). Harmonic Analysis in Phase Space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, New Jersey. |
[23] | M. Dimassi and J Sjortrand (1999) Spectral Asymptotic in Semi-Classical limit. London Mathematical Society Lecture Note Vol. 268 Cambridge University. |
[24] | C. Emmrich and A. Weinstein (1996) Geometry of the transport equation in multicomponent WKB approximation, Commun. Math. Phy. 176, 701-711. |
[25] | J. Bolte and R. Glaser (2004). Zitterbewegung and Semi-classical observables for Dirac Equation. arxiv; quant-Ph/0402154v/20Feb2004. |
APA Style
Ugwu, E. I. (2024). Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave. American Journal of Modern Physics, 13(3), 34-40. https://doi.org/10.11648/j.ajmp.20241303.11
ACS Style
Ugwu, E. I. Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave. Am. J. Mod. Phys. 2024, 13(3), 34-40. doi: 10.11648/j.ajmp.20241303.11
AMA Style
Ugwu EI. Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave. Am J Mod Phys. 2024;13(3):34-40. doi: 10.11648/j.ajmp.20241303.11
@article{10.11648/j.ajmp.20241303.11, author = {Emmanuel Ifeanyi Ugwu}, title = {Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave }, journal = {American Journal of Modern Physics}, volume = {13}, number = {3}, pages = {34-40}, doi = {10.11648/j.ajmp.20241303.11}, url = {https://doi.org/10.11648/j.ajmp.20241303.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20241303.11}, abstract = {The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion. }, year = {2024} }
TY - JOUR T1 - Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave AU - Emmanuel Ifeanyi Ugwu Y1 - 2024/07/29 PY - 2024 N1 - https://doi.org/10.11648/j.ajmp.20241303.11 DO - 10.11648/j.ajmp.20241303.11 T2 - American Journal of Modern Physics JF - American Journal of Modern Physics JO - American Journal of Modern Physics SP - 34 EP - 40 PB - Science Publishing Group SN - 2326-8891 UR - https://doi.org/10.11648/j.ajmp.20241303.11 AB - The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion. VL - 13 IS - 3 ER -