Skip to main content
SpringerLink
Log in

Saigo-Maeda Operators Involving the Appell Function, Real Spectra from Symmetric Quantum Hamiltonians and Violation of the Second Law of Thermodynamics for Quantum Damped Oscillators

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this study, we have generalized the fractional action integral by using the Saigo-Maeda fractional operators defined in terms of the Appell hypergeometric function of two variables , F3(a, a′, β, β′; γ; z, ζ) with complex parameters. We have derived the associated Euler-Lagrange equation and we have studied the harmonic oscillator problem. We have proved that a PT -symmetric quantum-mechanical Hamiltonian characterized by real and discrete spectra is obtained although the system is characterized by complex trajectories. The associated thermodynamical properties were discussed and it was revealed the entropy of the quantum system decreases with time toward an asymptotically positive value similar to what is observed in quantum Maxwell demon.

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

Similar content being viewed by others

Data Availability

The author confirms the absence of sharing data.

References

  1. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press (2012)

  2. A. B. Malinowska, T. Odzijewicz, D. F. M. Torres: Advanced Methods in the Fractional Calculus of Variations. SpringerBriefs in Applied Science and Engineering, Springer International Publishing; 2015

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

    MathSciNet  MATH  ADS  Google Scholar 

  4. El-Nabulsi, R.A., Torres, D.F.M.: Fractional action-like variational problems. J. Math. Phys. 49, 052521 (2008)

    Google Scholar 

  5. El-Nabulsi, R.A.: Fractional variational approach for dissipative mechanical systems. Anal. Theor. Appl. 30, 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  6. El-Nabulsi, R.A.: Path integral method for quantum dissipative systems with dynamical friction: Applications to quantum dots/zero-dimensional nanocrystals. Superlattices Microstruct. 144, 106581 (2020)

    Google Scholar 

  7. El-Nabulsi, R.A.: Fractional calculus of variations from extended Erdelyi-Kober operator. Int. J. Mod. Phys. B. 23, 3349–3361 (2009)

    MathSciNet  ADS  Google Scholar 

  8. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E53, 1890–1899 (1996)

    MathSciNet  ADS  Google Scholar 

  9. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E55, 3581–3592 (1997)

    MathSciNet  ADS  Google Scholar 

  10. Almeida, R., Malinowska, A.B., Torres, D.F.M.: A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51, 033503 (2010)

    MathSciNet  MATH  ADS  Google Scholar 

  11. El-Nabulsi, R.A.: Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator. Cent. Europ. J. Phys. 9, 250–256 (2011)

    ADS  Google Scholar 

  12. David, S.A., Valentim Jr., C.A.: Fractional Euler-Lagrange equations applied to oscillatory systems. Mathematics. 3, 269–272 (2015)

    MATH  Google Scholar 

  13. El-Nabulsi, R.A.: A fractional approach to nonconservative Lagrangian dynamical systems. Fiz. A. 14, 290–298 (2015)

    Google Scholar 

  14. A. Jajarmi, D. Baleanu, S. S. Sajjadi, J. H. Asad: A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Front. Phys.7. Article 196 (9 pages) (2018)

  15. R. Herrmann: Fractional Calculus: an Introduction for Physicists. World Scientific Publishing Company; (2011)

  16. K. Miller, B. Ross: An Introduction to Fractional Calculus and Fractional Differential Equations Wiley. New York; (1993)

  17. El-Nabulsi, R.A.: Fractional derivatives generalization of Einstein's field equations. Ind. J. Phys. 87, 195–200 (2013)

    Google Scholar 

  18. El-Nabulsi, R.A.: Gravitons in fractional action cosmology. Int. J. Theor. Phys. 51, 3978–3992 (2012)

    MathSciNet  MATH  Google Scholar 

  19. El-Nabulsi, R.A.: Fractional dynamics, fractional weak bosons masses and physics beyond the standard model, Chaos. Solitons & Fractals. 4, 2262–2270 (2009)

    Google Scholar 

  20. El-Nabulsi, R.A.: Modifications at large distances from fractional and fractal arguments. FRACTALS. 18, 185–190 (2010)

    MathSciNet  MATH  Google Scholar 

  21. El-Nabulsi, R.A.: Fractional functional with two occurrences of integrals and asymptotic optimal change of drift in the black-Scholes model. Acta Math. Viet. 40, 689–703 (2015)

    MathSciNet  MATH  Google Scholar 

  22. El-Nabulsi, R.A.: Complexified quantum field theory and “mass without mass” from multidimensional fractional actionlike variational approach with dynamical fractional exponents. Chaos, Solitons Fractals. 42, 2384–2398 (2009)

    MathSciNet  ADS  Google Scholar 

  23. El-Nabulsi, R.A.: Fractional Dirac operators and deformed field theory on Clifford algebra. Chaos, Solitons Fractals. 42, 2614–2622 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  24. El-Nabulsi, R.A.: Fractional field theories from multidimensional fractional variational problems. Int. J. Mod. Geom. Meth. Mod. Phys. 5, 863–892 (2008)

    MathSciNet  MATH  Google Scholar 

  25. El-Nabulsi, R.A., Wu, G.-c.: Fractional complexified field theory from Saxena-Kumbhat fractional integral, fractional derivative of order (α, β) and dynamical fractional integral exponent. African Disp. J. Math. 13, 45–61 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Saigo, M., Maeda, N.: More generalization of fractional calculus. Transf. Meth. Spec. Funct. 96, 386–400 (1996)

    MATH  Google Scholar 

  27. Slater, J.L.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  28. Carlson, B.C.: Lauricella’s hypergeometric function FD. J. Math. Anal. Appl. 7, 452–470 (1963)

    MathSciNet  MATH  Google Scholar 

  29. Humbert, P.: The confluent hypergeometric functions of two variables. P. Roy. Soc. Edinb. A41, 73–96 (1992)

    MATH  Google Scholar 

  30. Ancarani, L.U., Del Punta, J.A., Gasaneo, G.: Derivatives of horn hypergeometric functions with respect to their parameters. J. Math. Phys. 58, 073504 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  31. Colavecchia, F.D., Gasaneo, G., Garibotti, C.R.: Hypergeometric integrals arising in atomic collisions physics. J. Math. Phys. 38, 6603–6612 (1997)

    MathSciNet  MATH  ADS  Google Scholar 

  32. Agrawal, A.P., Choi, J., Jain, S.: Extended hypergeometric functions of two and three variables. Comm. Korean Math. Soc. 30(403–414), (2015)

  33. Wald, S., Henkel, M.: On integral representations and asymptotics of some hypergeometric functions in two variables. Int. Transf. Spec. Funct. 29, 95–112 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Vidunas, R.: Specialization of Appell’s functions to univariate hypergeometric functions. J. Math. Anal. Appl. 355, 145–163 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Brychkova, Y.A., Saad, N.: On some formulas for the Appell function F3(a,a′,b,b′;c;w,z). Int. Transf. Spec. Funct. 26, 910–923 (2015)

    Google Scholar 

  36. Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, New York (2007)

    MATH  Google Scholar 

  37. Saigo, M., Saxena, R.K., Ram, J.: On the two-dimensional generalized Weyl fractional calculus associated with two-dimensional H-transform. J. Frac. Calc. 8, 63–73 (1995)

    MathSciNet  MATH  Google Scholar 

  38. Saxena, R.K., Ram, J., Chandak, S.: On two-dimensional generalized Saigo fractional calculus associated with two-dimensional generalized H-transforms. J. Indian Acad. Math. 27, 167–180 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Saxena, R.K., Ram, J., Kumar, D.: Generalized fractional differentiation for Saigo operators involving aleph-function. J. Indian Acad. Math. 34, 109–115 (2012)

    MATH  Google Scholar 

  40. Saxena, R.K., Ram, J., Kumar, D.: Generalized fractional differentiation of the aleph-function associated with the Appell function F3. Acta Ciencia Indica. 38, 781–792 (2012)

    Google Scholar 

  41. Saxena, R.K., Ram, J., Suthar, D.L.: On two-dimensional Saigo-Maeda fractional calculus involving two-dimensional H-transforms. Acta Ciencia Indica. 30(813–822), (2014)

  42. Saxena, R.K., Saigo, M.: Generalized fractional calculus of the H-function associated with the Appell function F3. J. Fract. Calc. 19, 89–104 (2001)

    MathSciNet  MATH  Google Scholar 

  43. Srivastava, H.M., Saxena, R.K.: Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)

    MathSciNet  MATH  Google Scholar 

  44. Ibrahim, R.W.: An application of Lauricella hypergeometric functions to the generalized heat equations. Malaya J. Mat. 2, 43–48 (2014)

    MATH  Google Scholar 

  45. Bender, C.M., Holm, D.D., Hook, D.W.: Complexified dynamical systems. J. Phys. A Math. Theor. 40, F793–F804 (2007)

    MathSciNet  MATH  ADS  Google Scholar 

  46. Bender, C.M., Holm, D.D., Hook, D.W.: Complexified dynamical systems. J. Phys. A Math. Theor. 40, F81–F90 (2007)

    MathSciNet  MATH  ADS  Google Scholar 

  47. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonian having PT symmetry. Phys. Rev. Lett. 80(5243–5346), 5243–5246 (1998)

    MathSciNet  MATH  ADS  Google Scholar 

  48. Gosh, S., Modak, S.K.: Classical oscillator with position-dependent mass in a complex domain. Phys. Lett. A. 373, 1212–1217 (2009)

    MATH  ADS  Google Scholar 

  49. da Providencia, J., Bebiano, N., da Providencia, J.P.: Non-Hermitian Hamiltonians with real spectrum in quantum mechanics. Braz. J. Phys. 41, 78–85 (2011)

    MATH  ADS  Google Scholar 

  50. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    MathSciNet  ADS  Google Scholar 

  51. Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. 07, 1191–1306 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Graefe, E.M., Hoening, M., Korsch, H.J.: Classical limit of non-Hermitian quantum dynamics-a generalized canonical structure. J. Phys. A Math. Theor. 43, 7 (2010)

    MathSciNet  Google Scholar 

  53. El-Nabulsi, R.A.: Lagrangian and Hamiltonian dynamics with imaginary time. J. Appl. Anal. 18, 283–295 (2012)

    MathSciNet  MATH  Google Scholar 

  54. El-Nabulsi, R.A.: Quantization of non-standard Hamiltonians and the Riemann zeros. Qual. Theor. Dyn. Syst. 18, 69–84 (2019)

    MathSciNet  MATH  Google Scholar 

  55. Moiseyev, N.: Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 212–293 (1998)

    ADS  Google Scholar 

  56. K. Nigam, K. Banerjee, Quantum dynamics of complex Hamiltonians, arXiv: 1602.00157

  57. Bender, C.M.: Complex extension of quantum mechanics. Proc. Inst. Math. NAS Ukraine. 50, 617–628 (2004)

    MathSciNet  MATH  Google Scholar 

  58. Kaushal, R.S., Parthasarathi: Quantum mechanics of complex Hamiltonian systems in one dimension. J. Phys. A Math. Gen. 35, 8743–8761 (2002)

    MathSciNet  MATH  Google Scholar 

  59. Kaushal, R.S.: Classical and quantum mechanics of complex Hamiltonian systems: an extended complex phase space approach. Pramana J. Phys. 73, 287–297 (2009)

    ADS  Google Scholar 

  60. Ferreira, C., Lopez, J.L., Sinusia, E.P.: The third Appell function for one large variable. J. Approx. Theor. 165, 60–69 (2013)

    MathSciNet  MATH  Google Scholar 

  61. Vubangsi, M., Martin, T., Cornelius, F.L.: Harmonic oscillator with power-law increasing time-dependent effective mass. African Rev. Phys. 8, 341–347 (2013)

    Google Scholar 

  62. M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications; (1983)

  63. 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. B23, 3369–3386 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  64. El-Nabulsi, R.A., Soulati, T.A., Rezazadeh, H.: Non-standard complex Lagrangian dynamics. J. Adv. Res. Dyn. and Control Syst. 5, 50–62 (2013)

    MathSciNet  Google Scholar 

  65. El-Nabulsi, R.A.: Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace. Acta Math. Viet. 37, 149–160 (2012)

    MathSciNet  MATH  Google Scholar 

  66. El-Nabulsi, R.A.: Dirac operator of order 2/3 from Glaeske-Kilbas-Saigo fractional integral. Funct. Anal. Approx. Comp. 7, 15–28 (2015)

    Google Scholar 

  67. El-Nabulsi, R.A.: On nonlocal complexified Schrödinger equation and emergence of discrete quantum mechanics. Quant. Stud. Math. Found. 3, 327–335 (2016)

    MATH  Google Scholar 

  68. De Leo, S., Rodrigues, W.A.: Quantum mechanics: from complex to complexified quaternions. Int. J. Theor. Phys. 36, 2725–2757 (1997)

    MathSciNet  MATH  Google Scholar 

  69. Guralnik, G., Guralnik, Z.: Complexified path integrals and the phases of quantum field theory. Ann. Phys. 325, 2486–2498 (2010)

    MathSciNet  MATH  ADS  Google Scholar 

  70. Chandler, D.: Introduction to Modern Statistical Mechanics. Oxford University Press, Oxford (1987)

    Google Scholar 

  71. Lebedev, A.V., Oehri, D., Lesovik, G.B., Blatter, G.: Trading coherence and entropy by a quantum Maxwell demon. Phys. Rev. A94, 052133 (2016)

    ADS  Google Scholar 

  72. Yu, Y.: The second law of thermodynamics and entropy-decreasing processes with 4He superflows. Mod. Phys. Lett. B30, 1630008 (2016)

    ADS  Google Scholar 

  73. Prigogine, I., Stengers, I.: Order Out of Chaos. Bantam Books Inc., New York (1984)

    Google Scholar 

  74. Capek, V., Bok, J.: Violation of the second law of thermodynamics in the quantum microworld. Phys. A: Stat. Mech. Appl. 290, 379–401 (2001)

    MATH  Google Scholar 

  75. Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J., Evans, D.J.: Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89, 050601 (2002)

    ADS  Google Scholar 

  76. Ritort, F.: Work fluctuations, transient violations of the second law and free-energy recovery methods: perspectives in theory and experiments. Poincaré Seminar. 2, 193–277 (2003)

    Google Scholar 

  77. Jarzynski, C.: Equalities and inequalities: irreversibility and the second law of thermodynamics at the nanoscale. Annu. Rev. Condens. Matter Phys. 2, 329–351 (2011)

    ADS  Google Scholar 

  78. Seifert, U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012)

    ADS  Google Scholar 

  79. Martyushev, L.M.: Entropy and entropy production: old misconceptions and breakthroughs. Entropy. 15, 1152–1170 (2016)

    MathSciNet  MATH  ADS  Google Scholar 

  80. El-Nabulsi, R.A.: Nonlocal approach to nonequilibrium thermodynamics and nonlocal heat diffusion processes. Cont. Mech. Thermodyn. 30, 889–915 (2018)

    MathSciNet  MATH  Google Scholar 

  81. Boyacioglu, B., Chatterjee, A.: Heat capacity and entropy of a GaAs quantum dot with Gaussian confinement. J. Appl. Phys. 112, 083514 (2012)

    ADS  Google Scholar 

  82. Zubair, M., Mughal, M.J., Naqvi, Q.A.: An exact solution of spherical wave in D-dimensional fractional space. J. Electromagnet. Res. Appl. 25, 1481–1491 (2011)

    Google Scholar 

  83. Zubair, M., Mughal, M.J., Naqvi, Q.A.: The wave equation and general plane wave solutions in fractional space. Prog. Electromagnet. Res. Lett. 19, 137–146 (2010)

    Google Scholar 

  84. M. Zubair, M. J. Mughal, Q. A. Naqvi: Electromagnetic Wave Propagation in Fractional Space. In: Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer Briefs in Applied Sciences and Technology, Springer, Berlin, Heidelberg, (2012)

  85. Ostoja-Starzewski, M.: Continuum mechanics models of fractal porous media: integral relations and extremum principles. J. Mech. Mater. Struct. 4, 901–912 (2009)

    Google Scholar 

  86. Ostoja-Starzewski, M.: Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech. 205, 161–170 (2009)

    MATH  Google Scholar 

  87. Ostoja-Starzewski, M., Li, J.: Fractal materials, beams and fracture mechanics. Z. Ang. Math. Phys. 60, 1194–1205 (2009)

    MathSciNet  MATH  Google Scholar 

  88. Kamalov, T.F.: Axiomatization of mechanics. Quant. Comp. Comput. 11, 52–57 (2011)

    Google Scholar 

  89. Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A373, 1201–1211 (2009)

    MATH  ADS  Google Scholar 

Download references

Funding

The author received no direct funding for this work.

Author information

Authors and Affiliations

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

Disclosure of Conflicts of Interest

The author declares that he has no conflicts of interest.

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. Saigo-Maeda Operators Involving the Appell Function, Real Spectra from Symmetric Quantum Hamiltonians and Violation of the Second Law of Thermodynamics for Quantum Damped Oscillators. Int J Theor Phys 59, 3721–3736 (2020). https://doi.org/10.1007/s10773-020-04627-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-020-04627-6

Keywords

AMS Subject Classification

Search

Navigation