Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 083, 7 pages      arXiv:0908.1755      https://doi.org/10.3842/SIGMA.2009.083
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty

T.K. Jana a and P. Roy b
a) Department of Mathematics, R.S. Mahavidyalaya, Ghatal 721212, India
b) Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India

Received June 30, 2009, in final form August 10, 2009; Published online August 12, 2009

Abstract
We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in both the cases are found to be real. It is also shown that the models are η pseudo-Hermitian and the metric operator is found explicitly in both the cases.

Key words: non-Hermitian; minimal length.

pdf (207 kb)   ps (138 kb)   tex (10 kb)

References

  1. Kempf A., Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35 (1994), 4483-4496, hep-th/9311147.
  2. Kempf A., Mangano G., Mann R.B., Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995), 1108-1118, hep-th/9412167.
  3. Kempf A., Non-pointlike particles in harmonic oscillators, J. Phys. A: Math. Gen. 30 (1997), 2093-2101, hep-th/9604045.
  4. Hinrichsen H., Kempf A., Maximal localization in the presence of minimal uncertainties in positions and in momenta, J. Math. Phys. 37 (1996), 2121-2137.
  5. Garay L.J., Quantum gravity and minimum length, Internat. J. Modern Phys. A 10 (1995), 145-165, gr-qc/9403008.
  6. Gross D.J., Mende P.F., String theory beyond the Planck scale, Nuclear Phys. B 303 (1988), 407-454.
  7. Maggiore M., A generalized uncertainty principle in quantum gravity, Phys. Lett. B 304 (1993), 65-69, hep-th/9301067.
  8. Chang L.N., Minic D., Okamura N., Takeuchi T., Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations, Phys. Rev. D 65 (2002), 125027, 8 pages.
  9. Dadic I., Jonke L., Meljanac S., Harmonic oscillator with minimal length uncertainty relations and ladder operators, Phys. Rev. D 67 (2003), 087701, 4 pages, hep-th/0210264.
  10. Gemba K., Hlousek Z.T., Papp Z., Algebraic solution of the harmonic oscillator with minimal length uncertainty relations, arXiv:0712.2078.
  11. Brau F., Minimal length uncertainty relation and the hydrogen atom, J. Phys. A: Math. Gen. 32 (1999), 7691-7696, quant-ph/9905033.
  12. Fityo T.V., Vakarchuk I.O., Tkachuk V.M., One-dimensional Coulomb-like problem in deformed space with minimal length, J. Phys. A: Math. Gen. 39 (2006), 2143-2149, quant-ph/0507117.
  13. Akhoury R., Yao Y.-P., Minimal length uncertainty relation and the hydrogen spectrum, Phys. Lett. B 572 (2003), 37-42, hep-ph/0302108.
  14. Benczik S., Chang L.N., Minic D., Takeuchi T., Hydrogen-atom spectrum under a minimal-length hypothesis, Phys. Rev. A 72 (2005), 012104, 4 pages, hep-th/0502222.
  15. Nouicer K., Pauli-Hamiltonian in the presence of minimal lengths, J. Math. Phys. 47 (2006), 122102, 11 pages.
  16. Quesne C., Tkachuk V.M., Dirac oscillator with nonzero minimal uncertainty in position, J. Phys. A: Math. Gen. 38 (2005), 1747-1765, math-ph/0412052.
  17. Nouicer K., An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths, J. Phys. A: Math. Gen. 39 (2006), 5125-5134.
  18. Quesne C., Tkachuk V.M., Generalized deformed commutation relations with nonzero minimal uncertainties in position and/or momentum and applications to quantum mechanics, SIGMA 3 (2007), 016, 18 pages, quant-ph/0603077.
  19. Spector D., Minimal length uncertainty relations and new shape invariant models, J. Math. Phys. 49 (2008), 082101, 8 pages, arXiv:0707.1028.
  20. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
    Bender C.M., Boettcher S., Quasi-exactly solvable quartic potential, J. Phys. A: Math. Gen. 31 (1998), L273-L277, physics/9801007.
  21. Makris K.G., El-Ganainy R., Christodoulides D.N., Musslimani Z.H., Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100 (2008), 103904, 4 pages.
  22. Klaiman S., Günther U., Moiseyev N., Visualization of branch points in PT symmetric waveguides, Phys. Rev. Lett. 101 (2008), 080402, 4 pages, arXiv:0802.2457.
  23. Swanson M.S., Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys. 45 (2004), 585-601.
  24. Cooper F., Khare A., Sukhatme U.P., Supersymmetry in quantum mechanics, World Scientific, 2002.
  25. Mostafazadeh A., Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, math-ph/0107001.
    Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum J. Math. Phys. 43 (2002), 2814-2816, math-ph/0110016.
    Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry. III. Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (2002), 3944-3951, math-ph/0203005.
  26. Kamran N., Olver P.J., Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145 (1990), 342-356.
  27. Quesne C., Comment: "Application of nonlinear deformation algebra to a physical system with Pöschl-Teller potential" [Chen J.-L., Liu Y., Ge M.-L., J. Phys. A: Math. Gen. 31 (1998), 6473-6481], J. Phys. A: Math. Gen. 32 (1999), 6705-6710, math-ph/9911004.
    Chen J.-L., Liu Y., Ge M.-L., Application of nonlinear deformation algebra to a physical system with Pöschl-Teller potential, J. Phys. A: Math. Gen. 31 (1998), 6473-6481.

Previous article   Next article   Contents of Volume 5 (2009)