Abstract
In this study, we have constructed a generalized momentum operator based on the notion of backward–forward coordinates characterized by a low dynamical nonlocality decaying exponentially with position. We have derived the associated Schrödinger equation and we have studied the dynamics of a particle characterized by an exponentially decreasing position-dependent mass following the arguments of von Roos. In the absence of magnetic fields, it was observed that the dynamics of the particle is similar to the harmonic oscillator with damping and its energy state is affected by nonlocality. We have also studied the dynamics of a charged particle in the presence of Morse–Coulomb potentials and external magnetic and Aharonov-Bohm flux fields. Both the energy states and the thermodynamical properties were obtained. It was observed that all these physical quantities are affected by nonlocality and that for small magnetic fields and high quantum magnetic numbers, the entropy of the system decreases with increasing temperature unless the nonlocal parameter is negative. For positive value of the nonlocal parameter, it was found that the entropy increases with temperature and tends toward an asymptotically stable value similar to an isolated system.
Similar content being viewed by others
References
K. Nozari, A. Etemadi, Minimal length maximal momentum and Hilbert space representation of quantum mechanics. Phys. Rev. D 85, 104029 (2012)
S.M. Amirfakhrian, Spinless particle in a magnetic field under minimal length scenario. Z. Naturforsch. 71, 481–485 (2016)
B. Khosropour, Radiation and generalized uncertainty principle. Phys. Lett. B 785, 3–8 (2018)
S.K. Moayedi, M.R. Setare, B. Khosropour, Lagrangian formulation of a magnetostatic field in the presence of a minimal length scale based on the Kempf algebra. Int. J. Mod. Phys. A 28, 1350142 (2013)
R.A. El-Nabulsi, Generalized uncertainty principle in astrophysics from Fermi statistical physics arguments. Int. J. Theor. Phys. 59, 2083–2090 (2020)
R.A. El-Nabulsi, Some implications of three generalized uncertainty principles in statistical mechanics of an ideal gas. Eur. Phys. J. Plus 135, 34 (2020)
R.A. El-Nabulsi, On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. Roy. Soc. A476, 20190729 (2020)
F. Scardigli, The deformation parameter of the generalized uncertainty relation. J. Phys. Conf. Ser. 1275, 012004 (2019)
M. Izadparast, S.H. Mazharimousavi, Generalized extended momentum operator. Phys. Script. 95, 075220 (2020)
R.N. Costa Filho, J.P.M. Braga, J.H.S. Lira, J.S. Andrade Jr., Extended uncertainty from first principles. Phys. Lett. B 755, 367–370 (2016)
B. Hamil, M. Merad, Dirac and Klein–Gordon oscillators on anti-de Sitter space. Eur. Phys. J. Plus 133, 174 (2018)
B. Hamil, M. Merad, T. Birkandan, Applications of the extended uncertainty principle in AdS and dS spaces. Eur. Phys. J. Plus 134, 278 (2019)
W.S. Chung, H. Hassanabadi, Quantum mechanics on (anti)-de Sitter background II: Ramsauer–Townsend effect and WKB method. Mod. Phys. Lett. A 33, 1850150 (2018)
G.T. Einevoll, Operator ordering in effective mass theory for heterostructures II. strained systems. Phys. Rev. B 42, 3497 (1990)
P. Harrison, Quantum Wells, Wires and Dots (Wiley and Sons, New York, 2000)
J. Förster, A. Saenz, U. Wolff, Matrix algorithm for solving Schrödinger equations with position-dependent mass or complex optical potentials. Phys. Rev. E 86, 016701 (2012)
F.Q. Zhao, X.X. Liang, S.L. Ban, Influence of the spatially dependent effective mass on bound polarons in finite parabolic quantum wells. Eur. Phys. J. B 33, 3–8 (2003)
R.A. El-Nabulsi, Dirac equation with position-dependent mass and Coulomb-like field in Hausdorff dimension. Few Body Syst. 61, 1–10 (2020)
R.A. El-Nabulsi, A new approach to Schrodinger equation with position-dependent mass and its implications in quantum dots and semiconductors. J. Phys. Chem. Sol. 140, 109384 (2020)
R.A. El-Nabulsi, A generalized self-consistent approach to study position-dependent mass in semiconductors organic heterostructures and crystalline impure materials. Phys. E Low Dim. Syst. Nanostruct. 134, 114295 (2020)
E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1996)
R.A. El-Nabulsi, Dynamics of pulsatile flows through microtube from nonlocality. Mech. Res. Commun. 86, 18–26 (2017)
R.A. El-Nabulsi, Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics. Qual. Theor. Dyn. Syst. 16, 223–234 (2017)
R.A. El-Nabulsi, Modeling of electrical and mesoscopic circuits at quantum nanoscale from heat momentum operator. Phys. E Low-Dimensional Syst. Nanostruct. 98, 90–104 (2019)
R.A. El-Nabulsi, Nonlinear wave equations from a non-local complex backward-forward derivative operator. Waves Compl. Rand. Med. (2020). https://doi.org/10.1080/17455030.2019.1673502
R.A. El-Nabulsi, Massive photons in magnetic materials from nonlocal quantization. J. Magn. Magnet. Mat. 458, 213–216 (2018)
T.F. Kamalov, Classical and quantum-mechanical axioms with the higher time derivative formalism. J. Phys. Conf. Ser. 442, 012051 (2013)
T.F. Kamalov, Model of extended mechanics and non-local hidden variables for quantum theory. J. Russ. Laser Res. 30, 466–471 (2009). arxiv:0909.2678
T.F. Kamalov, Quantum corrections of Newton’s law of motion. Symmetry 12, 63 (2020)
J.A.K. Suykens, Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)
C.M. Bender, P.D. Mannheim, No-ghost theorem for the fourth-order derivative Pais–Uhlenbeck oscillator model Phys. Rev. Lett. 100, 110402 (2008)
J.D. Jackson, Classical Electrodynamics (John Wiley, New York, 1975)
J.Z. Simon, Higher Derivative Expansions and Non-Locality (University of California, Santa Barbara, August 1990). Ph.D. thesis
J.Z. Simon, Higher derivative Lagrangians. non-locality, problems, and solutions. Phys. Rev. D41, 3720 (1990)
S. Popescu, Dynamical quantum non-locality. Nat. Phys. 6, 151 (2010)
C. E. Pachon, L. A. Pachon, The origin of the dynamical quantum non-locality, arXiv: 1307.4144
B.C. da Costa, E.P. Borges, Generalized space and linear momentum operators in quantum mechanics. J. Math. Phys. 55, 062105 (2014)
X. Mei, P. Yu, The definition of universal momentum operator of quantum mechanics and the essence of micro-particle’s spin. J. Mod. Phys. 3, 451–470 (2012)
J. Li, M. Ostoja-Starzewski, Thermo-poromechanics of fractal media. Phil. Trans. R. Soc. A378, 20190288 (2020)
M. Ostoja-Starzewski, Electromagnetism on anisotropic fractal media. ZAMP 64, 381–390 (2013)
J. Li, M. Ostoja-Starzewski, Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A 465, 2521–2536 (2009)
M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J. Electromagn. Res. Appl. 25, 1481–1491 (2011)
M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog. Electromagn. Res. Lett. 19, 137–146 (2010)
R.A. El-Nabulsi, On generalized fractional spin, fractional angular momentum, fractional momentum operators in quantum mechanics. Few Body Syst. 61, 25 (2020)
O. Von Roos, Position-dependent effective mass in semiconductor theory. Phys. Rev. B 27, 7547 (1983)
O. Mustafa, Comment on ‘Two-dimensional position-dependent massive particles in the presence of magnetic fields’. J. Phys. A Math. Theor. 52, 148001 (2019)
J. Yu, S.-H. Dong, G.-H. Sun, Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential. Phys. Lett. A 322, 290–297 (2004)
S.H. Dong, J.J. Pena, C. Pacheco-Garcia, J. Garcia-Ravelo, Algebraic approach to the position-dependent mass Schrödinger for a singular oscillator. Mod. Phys. Lett. A 22, 1039–1045 (2007)
M. Eshghi, R. Sever, S.M. Ikhdair, Energy states of the Hulthén plus Coulomb-like potential with position-dependent mass function in external magnetic fields. Chin. Phys. B 27, 020301–5 (2018)
G. Ovando, J.J. Pena, J. Morales, J. Lopez-Bonilla, Position-dependent mass Schrödinger equation for exponential-type potentials. J. Mol. Model. 25, 289 (2019)
B. Gonul, B. Gonul, D. Tutcu, O. Ozer, Supersymmetric approach to exactly solvable systems with position-dependent effective masses. Mod. Phys. Lett. A 17, 2057–2066 (2002)
J. Bosse, Lorentz atom revisited by solving Abraham–Lorentz equation of motion. Z. Naturforsch. 72, 717–731 (2017)
T.G. Philbin, Quantum dynamics of the damped harmonic oscillator. New J. Phys. 14, 083043 (2012)
D.M. Gitman, V.G. Kupriyanov, The action principle for a system of differential equations. J. Phys. A Math. Theor. 40, 10071–10081 (2007)
M.C. Baldiotti, R. Fresneda, D.M. Gitman, Quantization of the damped harmonic oscillator revisited. Phys. Lett. A 375, 1630–1636 (2011)
M. Eshghi, R. Server, S.M. Ikhdair, Energy states of the Hulthén plus Coulomb-like potential with position-dependent mass function in external magnetic field. Chin. Phys. B 27, 020301 (2018)
M. Eshghi, H. Mehraban, S.M. Ikhdair, Approximate energy states and thermal properties of a particle with position-dependent mass in external magnetic fields. Chin. Phys. B 26(2017), 060302 (2017)
O. Mustafa, Z. Algadhi, Position-dependent mass charged particles in magnetic and Aharonov-Bohm flux field: separability, exact and conditionally exact solvability. Eur. Phys. J. P135, 559 (2020)
O. Mustafa, Z. Algadhi, Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality. Eur. Phys. J. P134, 228 (2019)
B.J. Falaye, G.-H. Sun, R. Silva-Ortigoza, S.-H. Dong, Hydrogen atom in a quantum plasma environment under the influence of Aharonov-Bohm flux and electric and magnetic fields. Phys. Rev. E 93, 053201 (2016)
M. Khosravi, B. Vasaghi, K. Abbasi, G. Rezaei, Magnetic susceptibility of cylindrical quantum dot with Aharonov-Bohm flux: simultaneous effects of pressure, temperature, and magnetic flux. J. Supercond. Novel Magnet. 33, 761–768 (2020)
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, Oxford, 1987)
M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing (Dover, New York, 1972)
O. Mustafa, PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian. Phys. Lett. A 384, 126265 (2020)
L.M. Martyushev, Entropy and entropy production: old misconceptions and breakthroughs. Entropy 15(2016), 1152–1170 (2016)
R.A. El-Nabulsi, Nonlocal approach to nonequilibrium thermodynamics and nonlocal heat diffusion processes. Cont. Mech. Thermodyn. 30, 889–915 (2018)
B. Boyacioglu, A. Chatterjee, Heat capacity and entropy of a GaAs quantum dot with Gaussian confinement. J. Appl. Phys. 112, 0 083514 (2012)
R.A. El-Nabulsi, Inverse-power potentials with positive-bounds energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments. Europ. Phys. J. P135, 693 (2020)
R.A. El-Nabulsi, Nonlocal-in-time kinetic energy description of superconductivity. Phys. C Supercond. Appl. 577, 1353716 (2020)
O. Mustafa, S. Habib Mazharimousavi, Ordering ambiguity revisited via position dependent mass pseudo-momentum operators. Int. J. Theor. Phys. 46, 1786–1796 (2007)
O. Mustafa, Comment on ’Nonlinear dynamics of a position-dependent mass-driven Duffing-type oscillator’. J. Phys. A Math. Theor. 46, 368001 (2013)
Acknowledgements
The author would like to thank the anonymous referee for his valuable comments which have improved this study.
Funding
The author received no direct funding for this work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The author declares that there is no conflict of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
El-Nabulsi, R.A. Nonlocal Thermodynamics Properties of Position-Dependent Mass Particle in Magnetic and Aharonov-Bohm Flux Fields. Few-Body Syst 61, 37 (2020). https://doi.org/10.1007/s00601-020-01569-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00601-020-01569-x