Skip to main content
SpringerLink
Log in

Position-dependent mass fractal Schrodinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this study, the Schrödinger equation with position-dependent mass in fractal dimensions is constructed from fractal anisotropy and product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media and elasticity. The theory is characterized by a fractal uncertainty relation and a generalized fractal momentum operator. The fractal Schrödinger equation is exactly solved for different position-dependent masses and effective potentials. In particular, we discuss the problems of quantum dots and nanocrystals. Modifications in their energies levels are detected which are in agreement with recent studies and experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

Data availability

The author confirms the absence of sharing data.

References

  • Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Appl. Math. Ser. 55, 1964 (1979)

    MATH  Google Scholar 

  • Adler, R.J., Chen, P., Santiago, D.I.: The Generalized uncertainty principle and black hole remnants. Gen. Rel. Grav. 33, 2101–2108 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Amirfakhrian, S.M.: Spinless particle in a magnetic field under minimal length scenario. Z. Naturforsch. 71, 481–485 (2016)

    Article  ADS  Google Scholar 

  • Baldereschi, A., Diaz, M.G.: Anisotropy of excitons in semiconductors. Nuovo Cimento B 1965–1970(68), 217–229 (1970)

    Article  ADS  Google Scholar 

  • Bhatti, M.: Fractional Schrödinger wave equation and fractional uncertainty principle. Int. J. Contemp. Math. Sci. 19(2), 943–950 (2007)

    Article  MATH  Google Scholar 

  • Bies, W.E., Radtke, R.J., Ehrenreich, H., Runge, E.: Thermoelectric properties of anisotropic semiconductors. Phys. Rev. B65, 085208 (2002)

    Article  ADS  Google Scholar 

  • Biswas, K., Saha, J.P., Patra, P.: On the position-dependent effective mass Hamiltonian. Eur. Phys. J. P135, 457 (2020)

    Google Scholar 

  • Bittle, E.G., Biacchi, A.J., Fredin, L.A., Herzing, A.A., Allison, T.C., Walker, A.R.H., Gundlach, D.J.: Correlating anisotropic mobility and intermolecular phonons in organic semiconductors to investigate transient localization. Commun. Phys. 2, 29 (2019)

    Article  Google Scholar 

  • Braidotti, M.C., Musslimani, Z.H., Conti, C.: Generalized uncertainty principle and analogue of quantum gravity in optics. Phys. D 338, 34–41 (2017)

    Article  Google Scholar 

  • Chatterjee, S., Valappil, N.V., Menon, V.M.: Investigation of steady-state and time-dependent luminescence properties of colloidal InGaP quantum dots. Mater. Res. Soc. Symp. Proc. 1133, 1133 (2009)

    Google Scholar 

  • Costa-Filho, R.N., Almeida, M.P., Farias, G.A., Andrade, J.S., Jr.: Displacement operator for quantum systems with position-dependent mass. Phys. Rev. A84, 050102 (2011)

    Article  ADS  Google Scholar 

  • Das, S., Vagenas, E.C.: Universality of quantum gravity corrections. Phys. Rev. Lett. 101, 221301 (2008)

    Article  ADS  Google Scholar 

  • Das, S., Vagenas, E.C.: Phenomenological implications of the generalized uncertainty principle. Can. J. Phys. 87, 233 (2009)

    Article  ADS  Google Scholar 

  • Das, S., Vagenas, E.C.: Reply to “Universality of quantum gravity corrections.” Phys. Rev. Lett. 104, 119002 (2010)

    Article  ADS  Google Scholar 

  • Demmie, P.N., Ostoja-Starzewski, M.: Waves in fractal media. J. Elast. 104, 187 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Dong, J.P., Xu, M.Y.: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 48, 072105 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dong, S.H., Ma, Z.Q., Esposito, G.: Exact solutions of the Schrödinger equation with inverse-power potential. Found. Phys. Lett. 12, 465–474 (1999)

    Article  MathSciNet  Google Scholar 

  • Dong, S.H., Pena, J.J., Pacheco-Garcia, C., Garcia-Ravelo, J.: Algebraic approach to the position-dependent mass Schrödinger for a singular oscillator. Mod. Phys. Lett. A 22, 1039–1045 (2007a)

    Article  ADS  MATH  Google Scholar 

  • Dong, S.-H., Pena, J.J., Pacheco-Garcia, C., Garcia-Revelo, J.: Algebraic approach to the position-dependent mass Schrödinger equation for a singular oscillator. Mod. Phys. Lett. A 22, 1039–1045 (2007b)

    Article  ADS  MATH  Google Scholar 

  • Dyatlov, S.: An introduction to fractal uncertainty principle. J. Math. Phys. 60, 081505 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dyatlov, S., Jin, J.: Resonances for open quantum maps and a fractal uncertainty principle. Commun. Math. Phys. 354, 269–316 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dyatlov, S., Zahl, J.: Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26, 1011–1094 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Dyatlov, S., Zworski, M.: Fractal uncertainty for transfer operators. Int. Math. Res. Not. 2020, 781–812 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Einevoll, G.T.: Operator ordering in effective mass theory for heterostructures II. Strained systems. Phys. Rev. B 42, 3497 (1990)

    Article  ADS  Google Scholar 

  • El-Nabulsi, R.A.: Fractional quantum Euler-Cauchy equation in the Schrödinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics. Mod. Phys. Lett. B 23, 3369–3386 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi, R.A.: Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors. Eur. Phys. J. plus 133, 394 (2018a)

    Article  Google Scholar 

  • El-Nabulsi, R.A.: Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J. Stat. Phys. 172, 1617–1640 (2018b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi, R.A.: Generalized uncertainty principle in astrophysics from Fermi statistical physics arguments. Int. J. Theor. Phys. 59, 2083–2090 (2020a)

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi, R.A.: Some implications of three generalized uncertainty principles in statistical mechanics of an ideal gas. Eur. Phys. J. P. 135, 34 (2020)

    Article  Google Scholar 

  • El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. R. Soc. A476, 20190729 (2020c)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi, R.A.: On generalized fractional spin, fractional angular momentum, fractional momentum operators in quantum mechanics. Few-Body Syst. 61, 25 (2020d)

    Article  ADS  Google Scholar 

  • El-Nabulsi, R.A.: Dirac equation with position-dependent mass and Coulomb-like field in Hausdorff dimension. Few Body Syst. 61, 1–10 (2020e)

    Article  ADS  Google Scholar 

  • El-Nabulsi, R.A.: 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)

    Article  Google Scholar 

  • El-Nabulsi, R.A.: 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)

    Article  Google Scholar 

  • El-Nabulsi, R.A.: Inverse-power potentials with positive-bounds energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments. Eur. Phys. J. P135, 693 (2020h)

    Google Scholar 

  • El-Nabulsi, R.A.: On nonlocal fractal laminar steady and unsteady flows. Acta Mech. 232, 1413–1424 (2021a)

    Article  MathSciNet  Google Scholar 

  • El-Nabulsi, R.A.: Dynamics of position-dependent mass particle in crystal lattices microstructures. Phys. E Low-Dim. Syst. Nanosystems. Nanostruct. 127, 114525 (2021)

    Article  Google Scholar 

  • Eshghi, M., Sever, R., Ikhdair, S.M.: Energy states of the Hulthén plus Coulomb-like potential with position-dependent mass function in external magnetic fields. Chin. Phys. B 27, 020301–020305 (2018)

    Article  ADS  Google Scholar 

  • Forstner, J., Weber, C., Danckwerts, J., Knorr, A.: Damping of electron density Rabi-oscillations and self-induced-transparency in semiconductor quantum dots. Postconf. Dig. Quantum Electron. Laser Sci. 5, 97 (2003). https://doi.org/10.1109/QELS.2003.238120

    Article  Google Scholar 

  • Fromhold, T.M.: Fractal resistance in a transistor. Nature 386, 124 (1997)

    Article  ADS  Google Scholar 

  • Garay, L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145–166 (1995)

    Article  ADS  Google Scholar 

  • Guillaumín-España, E., Núñez-Yépez, H.N., Salas-Brito, A.L.: Classical and quantum dynamics in an inverse square potential. J. Math. Phys. 55, 103509 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hamil, B., Merad, M., Birkandan, T.: Applications of the extended uncertainty principle in AdS and dS spaces. Eur. Phys. J. P. 134, 278 (2019)

    Article  Google Scholar 

  • Han, R., Schlag, W.: A higher-dimensional Bourgain-Dyatlov fractal uncertainty principle. Anal. PDE 13, 813–853 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison, P.: Quantum Wells, Wires and Dots. Wiley, New York (2000)

    Google Scholar 

  • Hegger, H., Huckestein, B., Hecker, K., Janssen, M., Freimuth, A., Reckziegel, G., Tuzinski, R.: Fractal conductance fluctuations in gold nanowires. Phys. Rev. Lett. 77, 3885–3888 (1996)

    Article  ADS  Google Scholar 

  • Ikhdair, S.M.: Effective Schrödinger equation with general ordering ambiguity position-dependent mass Morse potential. Mol. Phys. 110, 1415–1428 (2012)

    Article  ADS  Google Scholar 

  • Iomin, A.: Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos Solitons Fractals 44, 348–352 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ishkhanyan, A.M.: Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity. Europ. Phys. J. P133, 83 (2018)

    Google Scholar 

  • Izadparast, M., Mazharimousavi, S.H.: Generalized extended momentum operator. Phys. Script. 95, 075220 (2020)

    Article  Google Scholar 

  • Elechiguerra, J.L., Reyes-Gasga, J., Yacaman, M.J.: The role of twinning in shape evolution of anisotropic noble metal nanostructures. J. Mater. Chem. 16, 3906–3919 (2016)

    Article  Google Scholar 

  • Jahan, L., Boda, A., Chatterjee, A.: Ground state energy of an exciton in a spherical quantum dot in the presence of an external magnetic field. AIP Conf. Proc. 1661, 080008 (2015)

    Article  Google Scholar 

  • Jin, L., Zhang, R.: Fractal uncertainty principle with explicit exponent. Math. Ann. 376, 1031–1057 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Kamalov, T. F. Instability criterion and uncertainty relation, J. Phys. Conf. Ser. 1557(1), (2020) 012003.

  • Khosropour, B.: Radiation and generalized uncertainty principle. Phys. Lett. B 785, 3–8 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  • Kim, J., Baik, S.S., Ryu, S.H., Sohn, Y., Park, S., Park, B.-G., Denlinger, J., Yi, Y., Choi, H.J., Kim, K.S.: Observation of tunable band gap and anisotropic Dirac semimetal state in black phosphorus. Science 349, 723–726 (2015)

    Article  Google Scholar 

  • Landsman, N.P.: Lectures Notes on Hilbert Spaces and Quantum Mechanics, Lectures given at the Institute for Mathematics, Astrophysics, and Particle Physics. Radboud University Nijmegen, The Netherlands (2006)

    Google Scholar 

  • Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135–3145 (2000)

    Article  ADS  MATH  Google Scholar 

  • Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  • Lau, W., Singh, M.: The effect of anisotropy on the semiconductor to semimetal transition of type II semiconductor superlattices. Sol. State Commun. 100, 359–363 (1996)

    Article  ADS  Google Scholar 

  • Li, J., Ostoja-Starzewski, M.: Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A465, 2521 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Li, J., Ostoja-Starzewski, M.: Thermo-poromechanics of fractal media. Phil. Trans. R. Soc. A378, 20190288 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Luo, Z., Maassen, J., Deng, Y., Du, Y., Garrelts, R.P., Lundstrom, M.S., Ye, P.D., Xu, X.: Anisotropic in-plane thermal conductivity observed in few-layer black phosphorus. Nat. Commun. 6, 9572 (2015)

    Article  Google Scholar 

  • Malic, E., Bormann, M.J.P., Hovel, P., Kuntz, M., Bimberg, D., Knorr, M.A., Scholl, E.: Coulomb damped relaxation oscillations in semiconductor quantum dot lasers. IEEE J. Select. Topic Quant. Elect. 13, 1242–1248 (2007)

    Article  ADS  Google Scholar 

  • Micolich, A.P., Taylor, R.P., Newbury, R., Bird, J.P., Wirtz, R., Dettmann, C.P., Aoyagi, Y., Sugano, T.: Geometry induced fractal behavior in a semiconductor billiard. J. Phys. Condens. Matter 10, 1339–1347 (1998)

    Article  ADS  Google Scholar 

  • Micolich, A.P., Taylor, R.P., Newbury, R., Bird, J.P., Aoyagi, Y., Sugano, T.: Temperature dependence of the fractal dimension of magnetoconductance fluctuations in a mesoscopic semiconductor billiard. Superlatt. Microstruct. 25, 157–161 (1999)

    Article  ADS  Google Scholar 

  • Mukhopadhyay, M., Hazra, R.K., Ghosh, M., Mukherjee, S., Bhattacharyya, S.P.: Cent. Eur. J. Phys. 10, 983–988 (2012)

    Google Scholar 

  • Muslih, S.I., Agrawal, O.P., Baleanu, D.: A fractional Schrödinger equation and its solution. Int. J. Theor. Phys. 49, 1746–1752 (2010)

    Article  MATH  Google Scholar 

  • Mustafa, O., Algadhi, Z.: Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality. Eur. Phys. J. P134, 228 (2019)

    Google Scholar 

  • Mustafa, O., Mazharimousavi, S.H.: Ordering ambiguity revisited via position dependent mass pseudo-momentum operators. Int. J. Theor. Phys. 46, 1786–1796 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Mustard, D.: Uncertainty principles invariant under the fractional Fourier transform. J. Aust. Math. Soc. 33(2), 180–191 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3325–3339 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Ostoja-Starzewski, M.: Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech. 205, 161 (2009)

    Article  MATH  Google Scholar 

  • Ovondo, G., Pena, J.J., Morales, J., Lopez-Bonilla, J.: Position-dependent mass Schrödinger equation for the Morse potential. J. Phys. Conf. Ser. 792, 012037 (2017)

    Article  Google Scholar 

  • Ovondo, G., Pena, J.J., Morales, J., Lopez-Bonilla, J.: Position-dependent mass Schrödinger equation for exponential-type potentials. J. Mol. Model. 25, 289 (2019)

    Article  Google Scholar 

  • Palk, S.T.: Teaching renormalization, scaling, and universality with an example from quantum mechanics. J. Phys. Commun. 2, 015016 (2018)

    Article  Google Scholar 

  • Plato, A.D.K., Hughes, C.N., Kim, M.S.: Gravitational effects in quantum mechanics. Contemp. Phys. 57, 477–495 (2016)

    Article  ADS  Google Scholar 

  • Jin, R., Cao, Y.C., Hao, E., Métraux, G.S., Schatz, G.C., Mirkin, C.A.: Controlling anisotropic nanoparticle growth through plasmon excitation. Nature 425, 487–490 (2003)

    Article  ADS  Google Scholar 

  • Rego-Monteiro, M.A., Rodrigues, L.M.C.S., Curado, E.M.F.: Position-dependent mass quantum Hamiltonians: general approach and duality. J. Phys. A Math. Theor. 49, 125203 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Roy-Layinde, R.O., Vincent, U.E., Abolade, S.A., Papoola, O.O., Laoye, J.A., McClintock, P.V.E.: Vibrational resonances in driven oscillators with position-dependent mass. Phil. Trans. Roy. Soc. A 379, 20200227 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  • Sajanlal, P.R., Sreeprasad, T.S., Samal, A.K., Pradeep, T.: Anisotropic nanomaterials: structure, growth, assembly, and functions. Nano Rev. 2, 5883 (2011)

    Article  Google Scholar 

  • Sakurai, J.: Modern Quantum Mechanics. Addison-Wesley, London (1994)

    Google Scholar 

  • Scardigli, F.: The deformation parameter of the generalized uncertainty relation. J. Phys. Conf. Ser. 1275, 012004 (2019)

    Article  Google Scholar 

  • Schulze-Halberg, A.: Quasi-exactly solvable singular fractional power potentials emerging from the triconfluent Heun equation. Phys. Scr. 65, 373 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Shi, L., Yan, Z.: Effects of electric field and shape on the ground state energy of an exciton bound to an ionized donor impurity in ellipsoidal quantum dots. J. Appl. Phys. 114, 194301 (2013)

    Article  ADS  Google Scholar 

  • Shim, M., McDaniel, H.: Anisotropic nanocrystal heterostructures: Synthesis and lattice strain. Curr. Opin. Sol. State Mater. Sci. 14, 83–94 (2010)

    Article  ADS  Google Scholar 

  • Skara, F., Perivolaropoulos, L.: Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length in quantum mechanics. Phys. Rev. D 100, 123527 (2019)

    Article  ADS  Google Scholar 

  • Song, X.: An effective quark-antiquark potential for both heavy and light mesons. J. Phys. G Nucl. Part. Phys. 17, 49 (1991)

    Article  ADS  Google Scholar 

  • Taylor, R.P., Micolich, A.P., Newbury, R., Fromhold, T.M.: Correlation analysis of self-similarity in semiconductor billiards. Phys. Rev. B 56, R12733–R12756 (2010)

    Article  ADS  Google Scholar 

  • Taylor, R.P., Micolich, A.P., Newbury, R., Dettmann, C., Fromhold, T.M.: Fractal transistors. Semiconduct. Sci. Technol. 12, 1459–1464 (1997a)

    Article  ADS  Google Scholar 

  • Taylor, R.P., Newbury, R., Sachrajda, A.S., Feng, Y., Coleridge, P.T., Dettmann, C., Zhu, N., Guo, H., Delage, A., Kelly, P.J., Wasilewski, Z.: Self-similar magnetoresistance in a semiconductor Sinai billiard. Phys. Rev. Lett. 78, 1952–1955 (1997b)

    Article  ADS  Google Scholar 

  • Taylor, R.P., Micolich, A.P., Newbury, R., Bird, J.P., Fromhold, T.M., Cooper, J., Aoyagi, Y., Sugano, T.: Exact and statistical self-similarity in semiconductor billiards: a unified picture. Phys. Rev. B 58, 11107–11110 (1998)

    Article  ADS  Google Scholar 

  • Taylor, R.P., Micolich, A.P., Newbury, R., Fromhold, T.M., Tench, C.R.: Observation of fractal conductance fluctuations over three orders of magnitude. Aust. J. Phys. 52, 887–893 (1999)

    Article  ADS  Google Scholar 

  • Teschke, G.: Construction of generalized uncertainty principles and wavelets in anisotropic Sobolev spaces, Int. J. Wavelets. Multiresol. Inf. Process. 03, 189–209 (2005)

    Article  MATH  Google Scholar 

  • Von Roos, O.: Position-dependent effective mass in semiconductor theory. Phys. Rev. B 27, 7547 (1983)

    Article  ADS  Google Scholar 

  • Wójcik, D., Białynicki-Birula, I., Zyczkowski, K.: Time evolution of quantum fractals. Phys. Rev. Lett. 85, 5022–5025 (2000)

    Article  ADS  Google Scholar 

  • Yang, X.-J., Baleanu, D., Machado, J.A.T.: Mathematical aspects of the Heisenberg uncertainty principle with local fractional Fourier analysis. Bound. Val. Prob. 2013, 131 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, L., Zhou, Z., Song, J., Chen, X.: Anisotropic nanomaterials for shape-dependent physicochemical and biomedical applications. Chem. Soc. Rev. 48, 5140–5176 (2019)

    Article  Google Scholar 

  • Yu, J., Dong, S.-H., Sun, G.-H.: Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential. Phys. Lett. A 322, 290–297 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zhai, L.-X., Wang, Y., Liu, J.-J.: Exciton in an anisotropic parabolic quantum-well wire in the presence of a magnetic field. J. Appl. Phys. 110, 043701 (2011)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author would like to thank the group of anonymous referees for their useful comments and valuable suggestions.

Funding

The author would like to thank Chiang Mai University for funding this research.

Author information

Authors and Affiliations

  1. Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi

  2. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi

  3. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, 10671, Kolonaki, Athens, Greece

    Rami Ahmad El-Nabulsi

Authors
  1. Rami Ahmad El-Nabulsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Ethics declarations

Conflict of interest

The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Nabulsi, R.A. Position-dependent mass fractal Schrodinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals. Opt Quant Electron 53, 503 (2021). https://doi.org/10.1007/s11082-021-03093-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-021-03093-6

Keywords

PACS numbers

Search

Navigation