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Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

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Abstract.

The classical and quantum-mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics (i.e., \(p_{j} (\vec{x}) \rightarrow p_{j}(\vec{x}) -e A_{j}(\vec{x})\), where \(p_{j}(\vec{x})\) is the j-th component of the classical PDM-canonical-momentum), it admits a necessarily different and vital form in quantum mechanics (i.e., \(\hat{p}_{j}(\vec{x})/\sqrt{m(\vec{x})} \rightarrow (\hat{p}_{j}(\vec{x})\)\( -e A_{j}(\vec{x}))/\sqrt{m(\vec{x})}\), where \( \hat{p}_{j}(\vec{x})\) is the j-th component of the quantum PDM-momentum operator). Under our point transformation settings, only one of the two commonly used vector potentials (i.e., \(\vec{A} (\vec{x}) \sim (-x_{2}, x_{1}, 0)\)) is found eligible and is considered for our Illustrative examples.

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References

  1. O. von Roos, Phys. Rev. B 27, 7547 (1983)

    Article  ADS  Google Scholar 

  2. A. de Souza Dutra, C.A.S. Almeida, Phys. Lett. A 275, 25 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  3. R.A.C. Correa, A. de Souza Dutra, J.A. de Oliveira, M.G. Garcia, J. Math. Phys. 58, 012104 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  4. M.V. Loffe, D.N. Nishnianidze, V.V. Vereshagin, J. Math. Phys. 58, 072105 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  5. O. Mustafa, S.H. Mazharimousavi, Phys. Lett. A 358, 259 (2006)

    Article  ADS  Google Scholar 

  6. A.D. Alhaidari, Phys. Rev. A 66, 042116 (2002)

    Article  ADS  Google Scholar 

  7. B. Bagchi, A. Banerjee, C. Quesne, V.M. Tkachuk, J. Phys. A 38, 2929 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  8. O. Mustafa, S.H. Mazharimousavi, Phys. Lett. A 357, 295 (2006)

    Article  ADS  Google Scholar 

  9. O. Mustafa, J. Phys. A 44, 355303 (2011)

    Article  Google Scholar 

  10. B. Bagchi, P. Gorain, C. Quesne, R. Roychoudhury, Mod. Phys. Lett. A 19, 2765 (2004)

    Article  ADS  Google Scholar 

  11. A. de Souza Dutra, J.A. de Oliveira, J. Phys. A 42, 025304 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  12. S. Cruz y. Cruz, O. Rosas-Ortiz, J. Phys. A 42, 185205 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  13. S. Cruz y. Cruz, J. Negro, L.M. Nieto, Phys. Lett. A 369, 400 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. R. Koc, G. Sahinoglu, M. Koca, Eur. Phys. J. B 48, 583 (2005)

    Article  ADS  Google Scholar 

  15. S. Cruz y. Cruz, J. Negro, L.M. Nieto, J. Phys.: Conf. Ser. 128, 012053 (2008)

    Google Scholar 

  16. S. Ghosh, S.K. Modak, Phys. Lett. A 373, 1212 (2009)

    Article  ADS  Google Scholar 

  17. B. Bagchi, S. Das, S. Ghosh, S. Poria, J. Phys. A 46, 032001 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  18. S.H. Mazharimousevi, O. Mustafa, Phys. Scr. 87, 055008 (2013)

    Article  ADS  Google Scholar 

  19. P.M. Mathews, M. Lakshmanan, Quart. Appl. Math. 32, 215 (1974)

    Article  MathSciNet  Google Scholar 

  20. A. Venkatesan, M. Lakshmanan, Phys. Rev. E 55, 5134 (1997)

    Article  ADS  Google Scholar 

  21. J.F. Cariñena, M.F. Rañada, M. Santander, Regul. Chaotic Dyn. 10, 423 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  22. A. Bhuvaneswari, V.K. Chandrasekar, M. Santhilvelan, M. Lakshmanan, J. Math. Phys. 53, 073504 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  23. O. Mustafa, J. Phys. A 46, 368001 (2013)

    Article  MathSciNet  Google Scholar 

  24. A.K. Tiwari, S.N. Pandey, M. Santhilvelan, M. Lakshmanan, J. Math. Phys. 54, 053506 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  25. M. Lakshmanan, V.K. Chandrasekar, Eur. Phys. J. ST 222, 665 (2013)

    Article  Google Scholar 

  26. Z.E. Musielak, J. Phys. A 41, 055205 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  27. R. Bravo, M.S. Plyushchay, Phys. Rev. D 93, 105023 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  28. C. Quesne, J. Math. Phys. 59, 042104 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  29. C. Quesne, J. Math. Phys. 56, 012903 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  30. B.G. da Costa, E.P. Borges, J. Math. Phys. 59, 042101 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  31. C. Quesne, V.M. Tkachuk, J. Phys. A 37, 4267 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  32. B. Bagchi, A. Banerjee, C. Quesne, V.M. Tkachuk, J. Phys. A 38, 2929 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  33. J.F. Cariñena, M.F. Rañada, M. Santander, SIGMA 3, 030 (2007)

    Google Scholar 

  34. J.F. Cariñena, M.F. Rañada, M. Santander, Ann. Phys. 322, 434 (2007)

    Article  ADS  Google Scholar 

  35. J.F. Cariñena, M.F. Rañada, M. Santander, Rep. Math. Phys. 54, 285 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  36. C. Muriel, J.L. Romero, J. Phys. A 43, 434025 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  37. R.G. Pradeep, V.K. Chandrasekar, M. Santhilvelan, M. Lakshmanan, J. Math. Phys. 50, 052901 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  38. N. Euler, M. Euler, J. Nonlinear Math. Phys. 11, 399 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. K.S. Govinder, P.G.L. Leach, J. Math. Anal. Appl. 287, 399 (2003)

    Article  MathSciNet  Google Scholar 

  40. N. Euler, T. Wolf, P.G.L. Leach, M. Euler, Acta Appl. Math. 76, 89 (2003)

    Article  MathSciNet  Google Scholar 

  41. J.F. Cariñena, F.J. Herranz, M.F. Rañada, J. Math. Phys. 58, 022701 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  42. M. Ranada, Phys. Lett. A 379, 2267 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  43. O. Mustafa, J. Phys. A 48, 225206 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  44. M. Ranada, M.A. Rodrigues, M. Santander, J. Math. Phys. 51, 042901 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  45. M.A. Rodrigues, P. Tempesta, P. Winternitz, Phys. Rev. E 78, 046608 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  46. N.W. Evans, P.N. Verrier, J. Math. Phys. 49, 092902 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  47. M. Ranada, J. Math. Phys. 57, 052703 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  48. J.M. Jauch, E.L. Hill, Phys. Rev. 57, 641 (1940)

    Article  ADS  MathSciNet  Google Scholar 

  49. M. Ranada, J. Math. Phys. 38, 4165 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  50. F. Tremblay, A. Turbiner, P. Winternitz, J. Phys. A 43, 015202 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  51. W. Miller, S. Post, P. Winternitz, J. Phys. A 46, 423001 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  52. R.C.-Stursberg, J. Math. Phys. 55, 042904 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  53. O. Mustafa, S.H. Mazharimousavi, Int. J. Theor. Phys. 46, 1786 (2007)

    Article  Google Scholar 

  54. O. Mustafa, arXiv:1208.2109

  55. S. Gasirowicz, Quantum Mechanics, 3rd edition (John Wiley and Sons, Inc., New Jersey, 2003)

  56. O. Mustafa, S.H. Mazharimousavi, Int. J. Theor. Phys. 47, 1112 (2008)

    Article  Google Scholar 

  57. O. Mustafa, S.H. Mazharimousavi, J. Phys. A 40, 863 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  58. O. Mustafa, Int. J. Theor. Phys. 47, 1300 (2008)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Physics, Eastern Mediterranean University, G. Magusa, 10, Mersin, North Cyprus, Turkey

    Omar Mustafa & Zeinab Algadhi

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  1. Omar Mustafa
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  2. Zeinab Algadhi
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Correspondence to Omar Mustafa.

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Mustafa, O., Algadhi, Z. Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality. Eur. Phys. J. Plus 134, 228 (2019). https://doi.org/10.1140/epjp/i2019-12588-y

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