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Exact Solution of the Dirac Equation with a Combined Static Electric and Magnetic Field in the Context of Generalized Uncertainty Principle

There is no minimal uncertainty in position measurement in the Heisenberg uncertainty principle is to be considered as the minimum of space resolution, whereas numerous theories of quantum gravity predict the existence of a lower bound to the possible resolution of distances. The minimal length is considered commonly by a modification of the Heisenberg uncertainty principle into the generalized uncertainty principle (GUP). The application of GUP modifies every equation of motion of quantum mechanics and consequently, a new window of research has opened to study quantum mechanical problems under the framework of GUP. In this article, we present an exact solution of the Dirac equation with a combined static electric and magnetic field under the framework of GUP and obtain exact energy spectrums. The spectrums manifest a super-symmetry for the sufficient large magnetic field intensity compared to the electric field intensity. The methodology of the solution is designed for convenient implementation of the key property of the harmonic oscillator, the kinetic and potential energy parts of the Hamiltonian are of equal weight. An obligation for the existence of the solution is found that the magnetic field is stronger than the electric field. Our obtained result is confirmed by rendering energy levels of a relativistic electron in an external normal magnetic field, found in the literature.

Dirac Equation, Static Electric Field, Static Magnetic Field, Generalized Uncertainty Principle

APA Style

Md Moniruzzaman, Md Nasir Uddin, Syed Bodiuzzaman Faruque. (2022). Exact Solution of the Dirac Equation with a Combined Static Electric and Magnetic Field in the Context of Generalized Uncertainty Principle. International Journal of High Energy Physics, 9(2), 25-35. https://doi.org/10.11648/j.ijhep.20220902.11

ACS Style

Md Moniruzzaman; Md Nasir Uddin; Syed Bodiuzzaman Faruque. Exact Solution of the Dirac Equation with a Combined Static Electric and Magnetic Field in the Context of Generalized Uncertainty Principle. Int. J. High Energy Phys. 2022, 9(2), 25-35. doi: 10.11648/j.ijhep.20220902.11

AMA Style

Md Moniruzzaman, Md Nasir Uddin, Syed Bodiuzzaman Faruque. Exact Solution of the Dirac Equation with a Combined Static Electric and Magnetic Field in the Context of Generalized Uncertainty Principle. Int J High Energy Phys. 2022;9(2):25-35. doi: 10.11648/j.ijhep.20220902.11

Copyright © 2022 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. D. J. Gross and P. F. Mende, “The high-energy behavior of string scattering amplitudes,” Physics Letters B 197 (1987) 129-134.
2. D. Amati, M. Ciafaloni and G. Veneziano, “Superstring collisions at Planckian energies,” Physics Letters B 197 (1987) 81-88.
3. D. J. Gross and P. F. Mende, “String theory beyond the Planck scale,” Nuclear Physics B 303 (1988) 407-454.
4. D. Amati, M. Ciafaloni and G. Veneziano, “Classical and quantum gravity effects from Planckian energy superstring collisions,” International Journal of Modern Physics A 3 (1988) 1615-1661.
5. D. Amati, M. Ciafaloni and G. Veneziano, “Can spacetime be probed below the string size?,” Physics Letters B 216 (1989) 41-47.
6. E. Witten, “Reflections on the fate of spacetime,” Physics meets philosophy at the Planck scale (2001).
7. L. Susskind, “The world as a hologram,” Journal of Mathematical Physics 36 (1995) 6377-6396.
8. T. Padmanabhan, T. R. Seshadri and T. P. Singh, “Uncertainty principle and the quantum fluctuations of the schwarzschild light cones,” International Journal of Modern Physics A 1 (1986) 491-498.
9. R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. arXiv preprint hep-th/0109162 378 (2003) 207.
10. A. Kempf, “Uncertainty relation in quantum mechanics with quantum group symmetry,” Journal of Mathematical Physics 35 (1994) 4483-4496.
11. A. Kempf, G. Mangano and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Physical Review D 52 (1995) 1108.
12. H. Hinrichsen and A. Kempf, “Maximal localization in the presence of minimal uncertainties in positions and in momenta,” Journal of Mathematical Physics 37 (1996) 2121-2137.
13. A. Kempf, “Non-pointlike particles in harmonic oscillators,” Journal of Physics A: Mathematical and General 30 (1997) 2093.
14. J. Y. Bang and M. S. Berger, “Quantum mechanics and the generalized uncertainty principle,” Physical Review D 74 (2006) 125012.
15. L. J. Garay, “Models of neutrino masses and mixings,” Int. J. Mod. Phys. A 10 (1995) 145-166.
16. D. Bouaziz and M. Bawin, “Regularization of the singular inverse square potential in quantum mechanics with a minimal length,” Physical Review A 76 (2007) 032112.
17. M. Moniruzzaman and S. B. Faruque, “A Short Note on Minimal Length,” J. Sci. Res 11 (2019) 151.
18. M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,” Reviews of Modern Physics 73 (2001) 977.
19. A. W. Peet and J. Polchinski, “UV-IR relations in AdS dynamics,” Physical Review D 59 (1999) 065011.
20. K. Nozari, “Some aspects of Planck scale quantum optics,” Physics Letters B 629 (2005) 41-52.
21. L. N. Chang, D. Minic, N. Okamura and T. Takeuchi, “Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations,” Physical Review D 65 (2002) 125027.
22. K. Nozari and P. Pedram, “Minimal length and bouncing-particle spectrum,” EPL (Europhysics Letters) 92 (2010) 50013.
23. S. B. Faruque, M. A. Rahman and M. Moniruzzaman, “Upper bound on minimal length from deuteron,” Results in Physics 4 (2014) 52-53.
24. M. M. Stetsko, “Corrections to the n s levels of the hydrogen atom in deformed space with minimal length,” Physical Review A 74 (2006) 062105.
25. M. M. Stetsko and V. N. Tkachuk, “Scattering problem in deformed space with minimal length,” Physical Review A 76 (2007) 012707.
26. S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan et al, “Short distance versus long distance physics: The classical limit of the minimal length uncertainty relation,” Physical Review D 66 (2002) 026003.
27. A. Camacho, “Generalized uncertainty principle and quantum electrodynamics,” General Relativity and Gravitation 35 (2003) 1153-1160.
28. M. Moniruzzaman and S. B. Faruque, “Estimation of Minimal Length Using Binding Energy of Deuteron,” J. Sci. Res 10 (2018) 99.
29. M. V. Battisti and G. Montani, “Quantum dynamics of the Taub universe in a generalized uncertainty principle framework,” Physical Review D 77 (2008) 023518.
30. A. Boumali and H. Hassanabadi, “The exact solutions of a (2+1)-dimensional Dirac oscillator under a magnetic field in the presence of a minimal length,” Can. J. Phy 93 (2014) 542-548.
31. K. Nozari, “Generalized Dirac equation and its symmetries,” Chaos, Solitons & Fractals 32 (2007) 302-311.
32. K. Nouicer, “An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths,” Journal of Physics A: Mathematical and General 39 (2006) 5125.
33. C. Quesne and V. M. Tkachuk, “Dirac oscillator with nonzero minimal uncertainty in position,” Journal of Physics A: Mathematical and General 38 (2005) 1747.
34. M. S. Hossain and S. B. Faruque, “Influence of a generalized uncertainty principle on the energy spectrum of (1+1)-dimensional Dirac equation with linear potential,” Physica Scripta 78 (2008) 035006.
35. Y. Chargui, A. Trabelsi and L. Chetouani, “Exact solution of the (1+1)-dimensional Dirac equation with vector and scalar linear potentials in the presence of a minimal length,” Physics Letters A 374 (2010) 531-534.
36. M. Ara, M. Moniruzzaman and S. B. Faruque, “Exact solution of the Dirac equation with a linear potential under the influence of the generalized uncertainty principle,” Physica Scripta 82 (2010) 035005.
37. M. Moniruzzaman and S. B. Faruque, “The exact solution of the Dirac equation with a static magnetic field under the influence of the generalized uncertainty principle,” Physica Scripta 85 (2012) 035006. Corrigendum: “The exact solution of the Dirac equation with a static magnetic field under the influence of the generalized uncertainty principle,” Physica Scripta 86 (2012) 039503.
38. A. Iorio and P. Pais, “Generalized uncertainty principle in graphene,’’ J. Phys.: Conf. Ser. 1275 (2019) 012061.
39. P. Bosso, S. Das and V. Todorinov, “Quantum field theory with generalized uncertainty principle I: scalar electrodynamics,’’ Ann. Phys. 422 (2020) 168319.
40. P. Bosso and G. G. Luciano, “Generalized uncertainty principle: from the harmonic oscillator to a QFT toy model,’’ Eur. Phys. J. C. 81 (2021) 982.
41. E. Merzbacher, “Quantum Mechanics 2nd ed,” Wiely International Edition (1970).