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Pseudo-Invariant Approach for a Particle in a Complex Time-Dependent Linear Potential

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Abstract

The Lewis and Riesenfeld method is used by Ramos et al. (Eur. Phys. J. Plus 133, 449 2018) to study quantum systems governed by time-dependent \(\mathcal {P}\mathcal {T}\) symmetric Hamiltonians and particularly where the quantum system is a particle submitted to the action of a complex time-dependent linear potential. They have misleadingly claim that the \(\mathcal {P}\mathcal {T}\) invariant eigenstates are normalized with the Dirac delta function and obtained non physical observable expectation values of the position and the momentum operators x and p, as a consequence there is no uncertainty product for this system. We discuss and correct their results using the pseudo-invariant approach. To this end, we introduce a linear pseudo hermitian invariant operator which allows us to solve analytically the time-dependent Schrödinger equation for this problem and to construct a Gaussian wave packet solution. The normalization condition for the invariant eigenfunctions with the Dirac delta function is obtained. Then, using this Gaussian wave packet, we calculate the expectation values of the position and the momentum as well as the uncertainty product. We find that these expectation values of x and p are complex numbers but describe the classical motion. Also, we show that the uncertainty relation is physically acceptable.

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Notes

  1. The authors of Ref. [1] wrote (29), this means that eigenstates ϕλ(x,t) of the invariant operator I(t) are transformed under the \(\mathcal {P}\mathcal {T}\) action as \(\mathcal {P}\mathcal {T}{\phi }_{\lambda }(x,t)={\phi }_{\lambda }^{\ast }(-x,t).\) This implies that the eigenstates ϕλ(x,t) of I(t) are NOT eigenstates of the \(\mathcal {P}\mathcal {T}\) operator as claimed by the authors of Ref. [1]. One can see that in the following

    $$ \begin{array}{@{}rcl@{}} PT\phi_{\lambda}(x,t) &=& \sqrt{\frac{\sigma_{\lambda}}{2\pi b^{\ast}(t)}}\exp\left\{ \frac{-i}{b^{\ast}(t)}\left[ \left( \lambda-\gamma^{\ast}(t)\right) (-x)-\frac{a_{0}^{\ast}}{2}x^{2}\right] \right\} \\ &=& \sqrt{\frac{\sigma_{\lambda}}{-2\pi b(t)}}\exp\left\{ \frac{-i} {b(t)}\left[ \left( \lambda-c(t)\right) x-\frac{a_{0}}{2}x^{2}\right] \right\} \text{ \ }\\ && \neq\phi_{\lambda}(x,t). \end{array} $$
    (27)

    Morever these eigenstates of I(t) will not also satisfy the normalization condition with the Dirac delta function

    $$ {\int}_{-\infty}^{+\infty}\phi_{\lambda^{^{\prime}}}^{PT}(x,t)\phi_{\lambda }(x,t)dx=\frac{\sqrt{\sigma_{\lambda^{^{\prime}}}\sigma_{\lambda}}}{2\pi }\left[ \exp\frac{(\lambda-\lambda^{^{\prime}})}{\left\vert b(t)\right\vert }x\right]_{-\infty}^{+\infty} $$
    (28)

    where \(b(t)=i\left \vert b(t)\right \vert \) is purely imaginary.

References

  1. Ramos, B.F., Pedrosa, I.A., de Lima, A.L.: Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having \(\mathcal {P}\mathcal {T}\) symmetry. Eur. Phys. J. Plus 133, 449 (2018)

    Article  Google Scholar 

  2. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \(\mathcal {P}\mathcal {T}\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bender, C.M., Berntson, B., Parker, D., Samuel, E.: Observation of \(\mathcal {P}\mathcal {T}\) phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013)

    Article  ADS  Google Scholar 

  4. Rubinstein, J., Sternberg, P., Ma, Q.: Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. Phys. Rev. Lett. 99, 167003 (2007)

    Article  ADS  Google Scholar 

  5. Schindler, J., Lin, Z., Lee, J.M., Ramezani, H., Ellis, F.M., Kottos, T.: \(\mathcal {P}\mathcal {T}\) - symmetric electronics. J. Phys. A. 45, 444029 (2012)

    Article  ADS  MATH  Google Scholar 

  6. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in \(\mathcal {P}\mathcal {T}\) symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)

    Article  ADS  Google Scholar 

  7. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in \(\mathcal {P}\mathcal {T}\) periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)

    Article  ADS  Google Scholar 

  8. Feng, L., Wong, Z.J., Ma, R., Wang, Y., Zhang, X.: Single-mode laser by parity time symmetry breaking. Science 346, 972 (2014)

    Article  ADS  Google Scholar 

  9. Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D.N., Khajavikhan, M.: Parity-time-symmetric microring lasers. Science 346, 975 (2014)

    Article  ADS  Google Scholar 

  10. Feng, L., Xu, Y.-L., Fegadolli, W.G., Lu, M.-H., Oliveira, J.E.B., Almeida, V.R., Chen, Y.-F., Scherer, A.: Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. Nat. Matter 12, 108 (2013)

    Article  ADS  Google Scholar 

  11. Suchkov, S.V., Sukhorukov, A.A., Huang, J., Dmitriev, S.V., Lee, C., Kivshar, Y.S.: Nonlinear switching and solitons in \(\mathcal {P}\mathcal {T}\)-symmetric photonic systems. Laser Photonics Rev. 10, 177 (2016)

    Article  ADS  Google Scholar 

  12. Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  14. Bagchi, B., Quesne, C., Znojil, M.: Generalized continuity equation and modified normalization in \(\mathcal {P}\mathcal {T}\)-symmetric quantum mechanics. Mod. Phys. Lett. A 16, 2047 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Ahmed, Z.: Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex \(\mathcal {P}\mathcal {T}\)-invariant potential. Phys. Lett. A 282, 343 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Znojil, M.: Solvable simulation of a double-well problem in PT -symmetric quantum mechanics. J. Phys. A 36, 7639 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Weigert, S.: Completeness and orthonormality in \(\mathcal {P}\mathcal {T} \)-symmetric quantum systems. Phys. Rev. A 68, 062111 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  18. Ahmed, Z.: \(\mathcal {P}\)-, \(\mathcal {T}\)-, \(\mathcal {P}\mathcal {T}\)-, and \(\mathcal {C}\mathcal {P}\mathcal {T}\)-invariance of Hermitian Hamiltonians. Phys. Lett. A 310, 139 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Weigert, S.: Detecting broken \(\mathcal {P}\mathcal {T}\)-symmetry. J. Phys. A. 39, 10239 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Ahmed, Z.: Eigenvalue problems for the complex \(\mathcal {P}\mathcal {T} \)-symmetric potential V (x) = igx. Phys. Lett. A 364, 12 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  22. Scholtz, F.G., Geyer, H.B., Hahne, F.J.W.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. 213, 74–101 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Mostafazadeh, A.: Pseudo-Hermiticity versus \(\mathcal {P}\mathcal {T}\) symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Mostafazadeh, A.: Pseudo-Hermiticity versus \(\mathcal {P}\mathcal {T}\)-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum. J. Math. Phys. 43, 2814 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Mostafazadeh, A.: Pseudo-Hermiticity versus \(\mathcal {P}\mathcal {T} \)-symmetry III: equivalence of pseudo-Hermiticity and the presence of antilinear symmetries. J. Math. Phys. 43, 3944 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Dyson, F.J.: Thermodynamic behavior of an ideal ferromagnet. Phys. Rev. 102, 1230 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Figueira de Morisson Faria, C., Fring, A.: Time evolution of non-Hermitian Hamiltonian systems. J. Phys. A: Math. Theor. 39, 9269 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Figueira de Morisson Faria, C., Fring, A.: Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time-dependent quantum mechanical formulation. Laser Phys. 17, 424 (2007)

    Article  ADS  Google Scholar 

  30. Mostafazadeh, A.: Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator. Phys. Lett. B 650, 208 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Znojil, M.: Time-dependent version of crypto-Hermitian quantum theory. Phys. Rev. D 78, 085003 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  32. Znojil, M.: Three-Hilbert-space formulation of quantum mechanics. SIGMA 5, 001 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Znojil, M.: Crypto-unitary forms of quantum evolution operators. Int. J. Theor. Phys. 52, 2038 (2013)

    Article  MathSciNet  Google Scholar 

  34. Znojil, M.: Non-Hermitian Heisenberg representation. Phys. Lett. A 379, 2013 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Fring, A., Moussa, M.H.Y.: Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians. Phys. Rev. A 93, 042114 (2016)

    Article  ADS  Google Scholar 

  36. Fring, A., Moussa, M.H.Y.: Non-Hermitian Swanson model with a time-dependent metric. Phys. Rev. A 94, 042128 (2016)

    Article  ADS  Google Scholar 

  37. Miao, Y.-G., Xu, Z.-M.: Investigation of non-Hermitian Hamiltonians in the Heisenberg picture. Phys. Lett. A 380, 1805 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Luiz, F.S., Pontes, M.A., Moussa, M.H.Y.: Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps. arXiv:1611.08286 (2016)

  39. Fring, A., Frith, T.: Exact analytical solutions for timedependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians. Phys. Rev. A 95, 010102(R) (2017)

    Article  ADS  Google Scholar 

  40. Luiz, F.S., de Pontes, M.A., Moussa, M.H.Y.: Gauge linked time-dependent non-Hermitian Hamiltonians. arXiv:1703.01451 (2017)

  41. Maamache, M.: Non-unitary transformation of quantum time-dependent non-Hermitian systems. Acta Polytech. 57, 424 (2017)

    Article  Google Scholar 

  42. Znojil, M.: Non-Hermitian interaction representation and its use in relativistic quantum mechanics. Annals Phys. 385, 162 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Fring, A., Frith, T.: Time-dependent metric for the two-dimensional, non-Hermitian coupled oscillator. arXiv:1812.02862 (2018)

  44. Fring, A., Frith, T.: Solvable two-dimensional time dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT -regime. J. Phys. A 51, 265301 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Bagchi, B.: Evolution operator for time-dependent non-Hermitian Hamiltonians. Lett. High. Energy. Phys. 3, 04 (2018)

    Article  Google Scholar 

  46. Bíla, H.: Adiabatic time-dependent metrics in PT-symmetric quantum theories. arXiv:0902.0474 (2009)

  47. Gong, J., Wang, Q.H.: Geometric phase in PT -symmetric quantum mechanics. Phys. Rev. A 82, 012103 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  48. Gong, J., Wang, Q.H.: Timedependent PT -symmetric quantum mechanics. J. Phys. A 46, 485302 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gong, J., Wang, Q.-H.: Piecewise adiabatic following in non-Hermitian cycling. Phys. Rev. A. 97, 052126 (2018)

    Article  ADS  Google Scholar 

  50. Gong, J., Wang, Q.-H.: Piecewise adiabatic following: general analysis and exactly solvable models. Phys. Rev. A. 99, 012107 (2019)

    Article  ADS  Google Scholar 

  51. Zhang, D.-J., Wang, Q.-H., Gong, J.: Quantum geometric tensor in \(\mathcal {P}\mathcal {T}\)-symmetric quantum mechanics. Phys. Rev. A 99, 042104 (2019)

    Article  ADS  Google Scholar 

  52. Zhang, D.-J., Wang, Q.-H., Gong, J.: Time-dependent \(\mathcal {P},\mathcal {T}\)-symmetric quantum mechanics in generic non-Hermitian systems. arXiv:1906.03431 (2019)

  53. Maamache, M.: Periodic pseudo-Hermitian Hamiltonian: nonadiabatic geometric phase. Phys. Rev. A 92, 032106 (2015)

    Article  ADS  Google Scholar 

  54. Fring, A., Frith, T.: Mending the broken PT -regime via an explicit time-dependent Dyson map. Phys. Lett. A 381, 2318 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Fring, A., Frith, T.: Metric versus observable operator representation, higher spin models. Eur. Phys. J. 133, 57 (2018)

    Google Scholar 

  56. Fring, A., Frith, T.: Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians. Phys. Lett. A 383, 158 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. de Ponte, M.A., Luiz, F.S., Duarte, O.S., Moussa, M.H.Y.: All-creation and all-annihilation time-dependent \(\mathcal {P}\mathcal {T} \)-symmetric bosonic Hamiltonians: an infinite squeezing degree at a finite time. Phys. Rev. A. 100, 012128 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  58. Khantoul, B., Bounames, A., Maamache, M.: On the invariant method for the time-dependent non-Hermitian Hamiltonians. Eur. Phys. J. Plus 132, 258 (2017)

    Article  Google Scholar 

  59. Maamache, M., Djeghiour, O.-K., Mana, N., Koussa, W.: Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians. Eur. Phys. J. Plus 132, 383 (2017)

    Article  MATH  Google Scholar 

  60. Koussa, W., Mana, N., Djeghiour, O.-K., Maamache, M.: The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. J. Math. Phys. 59, 072103 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Lewis, H.R., Riesenfeld, W.B.: An exact quantum theory of the time dependent harmonic oscillator and of a charged particle time dependent electromagnetic field. J. Math. Phys. 10, 1458 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Wigner, E.: Group Theory and its Application to Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959)

    MATH  Google Scholar 

  63. Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Quantum Electrodynamics. Pergamon Press, Oxford (1982)

    Google Scholar 

  64. Chern, B., Tubis, A.: Invariance principles in classical and quantum mechanics. Am. J. Phys. 35, 254 (1967)

    Article  ADS  Google Scholar 

  65. de Sousa Dutra, A., Hott, M.B., dos Santos, V.G.C.S.: Time-dependent non-Hermitian Hamiltonians with real energies. Europhys. Lett. 71, 166 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  66. Yuce, C.: Time-dependent PT-symmetric problems. Phys. Lett. A 336, 290 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. Yuce, C.: Complex spectrum of a spontaneously unbroken PT symmetric hamiltonian. arXiv:0703235v1 (2007)

  68. Moiseyev, N.: Crossing rule for a PT -symmetric two-level time-periodic system. Phys. Rev. A 83, 052125 (2011)

    Article  ADS  Google Scholar 

  69. Luo, X., Huang, J., Zhong, H., Qin, X., Xie, Q., Kivshar, Y.S., Lee, C.: Pseudo-parity-time symmetry in optical systems. Phys. Rev. Lett. 110, 243902 (2013)

    Article  ADS  Google Scholar 

  70. Luo, X., Wu, D., Luo, S., Guo, Y., Yu, X., Hu, Q.: Pseudo-parity–time symmetry in periodically high-frequency driven systems: perturbative analysis. J. Phys. A 47, 345301 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  71. Maamache, M., Lamri, S., Cherbal, O.: Pseudo PT-symmetry in time periodic non-Hermitian Hamiltonians systems. Annals Phys. 378, 150 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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  1. Laboratoire de Physique Quantique et Systèmes Dynamiques, Faculté des Sciences, Université Ferhat Abbas Sétif 1, Sétif, 19000, Algeria

    Walid Koussa & Mustapha Maamache

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  1. Walid Koussa
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Koussa, W., Maamache, M. Pseudo-Invariant Approach for a Particle in a Complex Time-Dependent Linear Potential. Int J Theor Phys 59, 1490–1503 (2020). https://doi.org/10.1007/s10773-020-04417-0

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