[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Integrability of a Time Dependent Coupled Harmonic Oscillator in Higher Dimensions

Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 81--94 | DOI:10.5890/DNC.2018.03.007

Ram Mehar Singh$^{1}$, S B Bhardwaj$^{2}$, Kushal Sharma$^{3}$, Richa Rani$^{2}$, Fakir Chand$^{2}$, Anand Malik$^{4}$

$^{1}$ Department of Physics, Chaudhary Devi Lal University Sirsa-125055, India

$^{2}$ Department of Physics, Kurukshetra University Kurukshetra-136119, India

$^{3}$ Department of Mathematics, National Institute of Technology, Hamirpur-177005, India

$^{4}$ Department of Physics, Chaudhary Bansi Lal University, Bhiwani-127021, India

Download Full Text PDF

 

Abstract

Within the frame work of extended complex phase space characterized by x = x1 + ip4,y = x2 + ip5,z = x3 + ip6, px = p1 + ix4, py = p2 + ix5 and pz = p3 +ix6, we investigate the exact dynamical invariant for a coupled harmonic system in three dimensions. For this purpose Lie-algebraic method is employed and the invariant obtained in this work may play an important role in reducing the order of differential equations, solution of Cauchy system and to check the accuracy of a numerical simulation.

Acknowledgments

The authors are thankful to the learned referees for several useful comments which helped in considerably improving and fine-tuning some of the ideas in the original version of the paper.

References

  1. [1]  Kaushal, R.S. (1998), Classical and Quantum Mechanics of Noncentral Potentials, Narosa Publishing House, New Delhi.
  2. [2]  Singh, R.M., Bhardwaj, S.B. and Mishra, S.C. (2013), Closed-form solutions of the Schrödinger equation for a coupled harmonic potential in three dimensions, Computers & Math. with Appl., 66, 537-541. Singh, R.M., Chand, F. and Mishra, S.C. (2009), The solution of the Schrödinger equation for coupled quadratic and quartic potentials in two dimensions, Pramana J Phys., 72, 647-654.
  3. [3]  Khare, A. and Mandal, B.P. (2000), A PT-invariant potential with complex QES eigenvalues, Phys. Lett. A 272, 53-56.
  4. [4]  Singh, S. and Kaushal, R.S. (2003), Complex dynamical invariants for one-dimensional classical systems, Phys. Scr. 67, 181-185.
  5. [5]  Bender, C.M. (2007), Making sense of non-Hermitian Hamiltonians, Reports on Progress in Physics, 70, 947-1018.
  6. [6]  Kaushal, R.S. and Korsch, H.J. (2000), Some remarks on complex Hamiltonian systems, Phys. Lett. A, 276, 47-51.
  7. [7]  Nelson, D.R. and Snerb, N.M. (1998), Non-Hermitian localization and population biology, Phys. Rev. E, 58, 1383- 1403.
  8. [8]  Hatano, N. and Nelson, D.R. (1997), Vortex depinning and non-Hermitian quantum mechanics, Phys. Rev., 56, 8651- 8673.
  9. [9]  Xavier, Jr. A.L. and de Aguiar, M.A.M. (1996), Complex trajectories in the quartic oscillator and its semiclassical coherent state, Ann. Phys.(N.Y.), 252, 458-478.
  10. [10]  Bender, C.M. , Boettcher, S. and Meisinger, P.N. (1999), PT-symmetric quantum mechanics, J. Math. Phys., 40, 2201- 2229.
  11. [11]  Bhardwaj, S.B. , Singh, R.M. and Mishra, S.C.(2014), Eigenspectra of a complex coupled harmonic potential in three dimensions, Comp. & Math. Appl., 68, 2068-2079.
  12. [12]  Bhardwaj, S.B. , Singh, R.M. and Mishra, S.C.(2016), Quantum mechanics of PT and non-PT -symmetric potentials in three dimensions, Pramana-J Phys., 87, 1-10.
  13. [13]  Parthasarthi and Kaushal, R.S.(2003), Quantum mechanics of complex sextic potentials in one dimension, Phys Scr., 68, 115-127.
  14. [14]  Bhardwaj, S.B. , Singh, R.M., Mishra , S.C. and Sharma, K. (2015), On Solving the Schrödinger Equation for Three- Dimensional Noncentral Potential, J. Adv. Phys., 4, 215-218. Bhardwaj, S.B. and Singh, R.M. (2016), Exact solutions of 3-dimensional Schrödinger equation with a coupled quartic potential, J. Adv. Phys., 5, 44-46.
  15. [15]  Bender, C.M. and Turbiner, A. (1993), Analytic continuation of eigenvalue problems, Phys. Lett. A, 173, 442-446.
  16. [16]  Hietarinta, J. (1987), Direct methods for the search of the second invariant, Phys. Rep. 147, 87-154.
  17. [17]  Whittaker, E.T. (1960), A treatise on the analytical dynamics of particle and rigid bodies, Cambridge University press, London, p246.
  18. [18]  Singh, R.M.(2015), Integrability of a coupled harmonic oscillator in extended complex phase space, Discontinuity, Nonlinearity and Complexity, 4, 35-48. Rao N.N., Buti B. and Khadkikar S.B.(1986), Hamiltonian systems with indefinite kinetic energy, Pramana J. Phys., 27, 497-505.
  19. [19]  Colgerave, R.K., Croxson, P., and Mannan, M.A.(1988), Complex invariants for the time-dependent harmonic oscillator, Phys. Lett. A, 131, 407-410.
  20. [20]  Moiseyev, N.(1998), Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Phys.Rep., 302, 212-293. Virdi J.S., Chand F. , Kumar C.N., and Mishra S.C. (2012), Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems, Can. J. Phys., 90, 151-157.
  21. [21]  Whiteman, K.J. (1977), Invariants and stability in classical mechanics, Rep. Prog. Phys., 40, 1033-1069.
  22. [22]  Kusenko, A. and Shrock, R. (1994), General determination of phases in quark mass matrices, Phys. Rev. D, 50, R30- R33.
  23. [23]  Gunion, J.F. and Haber, H.E.(2005), Conditions for CP-violation in the general two-Higgs-doublet model, Phys. Rev. D, 72, 095002-21 .
  24. [24]  Plebanski, J.F. and Demianski, M.(1976), Rotating, charged, and uniformly accelerating mass in general relativity, Ann. Phys.(N.Y.), 98, 98-127.
  25. [25]  Kumar, C.N. and Khare, A.(1989), Chaos in gauge theories possessing vortices and monopole solutions, J. Phys. A: Math.& Gen., 22, L849-L853.
  26. [26]  Savidy G.K. (1983), The Yang-Mills classical mechanics as a Kolmogorov K-system, Phys. Lett. B, L30, 303-307.
  27. [27]  Virdi, J.S., Chand, F., Kumar, C.N. and Mishra, S.C. (2012), Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems, Can. J. Phys., 90, 151-157. Mishra, S.C. and Chand, F. (2006), Construction of exact dynaical invariants of two-dimensional classical system, Pramana J. Phys., 66, 601-607.
  28. [28]  Chand, F., Kumar, N. and Mishra, S.C. (2015), Exact fourth order invariants for one-dimensional time-dependent Hamiltonian systems, Indian J. Phys., 89, 709-712.
  29. [29]  Kaushal, R.S. and Singh, S. (2001), Construction of complex invariants for classical dynamical systems, Ann. Phys. (N.Y.), 288, 253-276.
  30. [30]  Bhardwaj, S.B., Singh, R.M. and Sharma, K. (2017), Complex dynamical invariants for a PT - symmetric Hamiltonian system in higher dimensions, Chinese J. Phys., (55), 533-542.
  31. [31]  Tabour,M. (1989), Chaos and Integrability in Nonlinear Dynamics,Wiley publications, New York .

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates