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The time independent fractional Schrödinger equation with position-dependent mass. (English) Zbl 07464417

Summary: In this paper, we present a fractional form of Schrödinger equation using the Kallil’s derivative method. Then we investigate the capability of this equation to obtain the wave functions and its energy levels for a particle trapped in infinite potential well (IPW) for the range \(1 < \alpha \leq 2\). We also, present the fractional form of the Schrödinger equation for a particle with position-dependent mass (PDM) and by using it, we obtain wave functions and its energy levels in range \(0 < \alpha\leq 1\) and in different amounts of \(\eta\) and \(\mu\) in which \(\mu + \eta = 1/2\). It seems that our method is very useful to solve the problems in PDM case.

MSC:

82-XX Statistical mechanics, structure of matter
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[1] Yosida, K., Functional Analysis (1970), Springer-Verlag Berlin Heidelberg · Zbl 0217.16001
[2] Wei, Y., The infinite square well problem in the standard, fractional, and relativistic quantum mechanics, Int. J. Theor. Math. Phys., 5, 4, 58-65 (2015)
[3] Chung, W. S.; Zare, S.; Hassanabadi, H., Investigation of conformable fractional Schrödinger equation in presence of killingbeck and hyperbolic potentials, Common. Theor. Phys., 67, 250-254 (2017) · Zbl 1360.81168
[4] Tarasov, V., Quantum Mechanics of Non-Hamiltonian and Dissipative Systems (2008), Elsevier · Zbl 1213.81004
[5] Herrman, R., Fractional Calculus an Introduction for Physicists (2014), World Scientific Publishing · Zbl 1293.26001
[6] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, Article 056108 pp. (2002)
[7] Naber, M., Time fractional Schrödinger equation, J. Math. Phys., 45, 3339 (2004) · Zbl 1071.81035
[8] Al-Raeei, M.; Sayem El-Daher, M., A numerical method for fractional Schrödinger equation of Lennard-Jones potential, Phys. Lett. A, 25831 (2019) · Zbl 1476.81032
[9] Liu, N.; Jiang, W., A numerical method for solving the time fractional Schrödinger equation, (Mathematics Subject Classification (2010))
[10] Tarasov, V. E., Fractional Heisenberg equation, Phys. Lett. A, 372 (2006)
[11] Tarasov, V. E., Fractional Heisenberg equation, Phys. Lett. A, 372 (2008) · Zbl 1220.81097
[12] Rabei, E. M.; Tarawneh, D.; Muslih, S. I.; Baleanu, D., Heisenberg’s equations of motion with fractional derivatives, J. Vib. Control, 13, 9-10, 1239-1247 (2007) · Zbl 1161.81352
[13] Tarasov, V. E., Quantum dissipation from power-law memory, Ann. Physics, 327 (2012) · Zbl 1246.81089
[14] Tarasov, V. E., Fractional derivative as fractional power of derivative, Internat. J. Math., 18, 3 (2007) · Zbl 1119.26011
[15] Wang, S.; Xu, M., Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys., 48, Article 043502 pp. (2007) · Zbl 1137.81328
[16] Balakrishnan, V., Fractional power of closed operator and the semi-group generated by them, Pacific J. Math., 10, 419-437 (1960) · Zbl 0103.33502
[17] Bochner, S., Diffusion equations and stochastic processes, Proc. Nat. Acad. Sci USA, 35, 343-396 (1949)
[18] Tarasov, V. E., Fractional generalization of quantum Markovian master equation, Theoret. Math. Phys., 158 (2008) · Zbl 1176.81070
[19] Ginibrjz, J.; Velo, G., On a class of nonlinear Schrödinger equations I. The Cauchy problem, general case, J. Funct. Anal., 32, l-32 (1979) · Zbl 0396.35028
[20] Herrmann, R., Properties of a fractional derivative Schrödinger type wave equation and a new interpretation of the charmonium spectrum (2006), arXiv:math-ph:0510099
[21] Morita, T.; Sato, K. I., Liouville and Riemann-Liouville fractional derivatives via contour integrals, Int. J. Theory Appl., 16, 3 (2013) · Zbl 1312.30051
[22] Madrid, Y.; Molina, M.; Torres, R., Quantum Fractional Fourier Transform (2018), Optical Society of America, paper JTu2A.73
[23] Cui-Hong, L.; Hong-Yi, F.; Dong-Wei, L., From fractional Fourier transformation to quantum mechanical fractional squeezing transformation, Chin. Phys. B, 24, 2 (2015)
[24] Khalili. Golmankhaneh, A. R.; Golmankhaneh, A. K.; Baleanu, D., On nonlinear fractional Klein-Gordon equation, Signal Process., 91, 446-451 (2011) · Zbl 1203.94031
[25] Baleanu, D.; Golmankhaneh, A. R.; Golmankhaneh, A. K., Rom. J. Phys., 54, 9-10, 823-832 (2009) · Zbl 1231.65188
[26] Khalil, R.; Al Horani, M.; Yousef, A.; Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014) · Zbl 1297.26013
[27] El-Nabulsi, R. A., Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics, Eur. Phys. J. Plus, 134, 192 (2019)
[28] von Roos, Oldwig, Position-dependent effective masses in semiconductor theory, Phys. Rev. B, 27, 12 (1983)
[29] Bastard, G., (Wave Mechanics Applied To Semiconductor Heterostructures. Wave Mechanics Applied To Semiconductor Heterostructures, Editions de Physique (1998), Les Ulis)
[30] Ring, P.; Schuck, P., The Nuclear Many Body Problem (1980), Springer: Springer NewYork
[31] Serra, L.; Lipparini, E., Europhys. Lett., 40, 667 (1997)
[32] Cruz y. Cruz, S.; Negro, J.; Nieto, L. M., Classical and quantum position-dependent mass harmonic oscillators, Phys. Lett. A, 369, 400-406 (2007) · Zbl 1209.81095
[33] Mustafa, O., Two-dimensional position-dependent mass Lagrangians Superintegrability and exact solvability (2013)
[34] Mustafa, O.; Mazharimousavi, S. H., A singular position-dependent mass particle in an infinite potential well, Phys. Lett. A, 373, 325-327 (2009) · Zbl 1227.81166
[35] Zare, S.; Hassanabadi, H., Properties of quasi-oscillator in PDM formalism, Adv. High Energy Phys. (2016), Article ID 4717012 · Zbl 1366.81161
[36] Ovando, G.; Peña, J. J.; Morales, J., Position-dependent mass Schrödinger equation for the morse potential, IOP Conf. Series: J. Phys.: Conf. Ser., 792, Article 012037 pp. (2017)
[37] Henini, M., Molecular Beam Epitaxy (2018), Elsevier, (chapters 30-31)
[38] von Roos, O., Position-dependent effective masses in semiconductor theory, Phys. Rev. B, 27, 12 (1983)
[39] da Costa, B. G.; Borges, E. P., A position-dependent mass harmonic oscillator and deformed space, J. Math. Phys., 59, Article 042101 pp. (2018) · Zbl 1386.81100
[40] Laskin, N., Fractional Quantum Mechanics (2018), World Scientific Publishing · Zbl 1425.81007
[41] Al-Raeeia, M.; El-Daher, M. S., Numerical simulation of the space dependent fractional Schrödinger equation for London dispersion potential type, Heliyon, 6, 7, Article e0449 pp. (2020)
[42] Ikot, A. N.; Awoga, O. A.; Antia, A. D.; Hassanabadi, H.; Maghsoodi, E., Approximate solutions of D-dimensional Klein-Gordon equation with modified hylleraas potential, Few-Body Syst., 54, 11, 2041-2051 (2013)
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