Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty

T. K. Jana ; P. Roy

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

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 $\eta$ pseudo-Hermitian and the metric operator is found explicitly in both the cases.

Détail

Keywords: non-Hermitian; minimal length.

Bibliographie

[1] Kempf A., “Uncertainty relation in quantum mechanics with quantum group symmetry”, J. Math. Phys., 35 (1994), 4483–4496 ; hep-th/9311147 | DOI | MR | Zbl

[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 | DOI | MR

[3] Kempf A., “Non-pointlike particles in harmonic oscillators”, J. Phys. A: Math. Gen., 30 (1997), 2093–2101 ; hep-th/9604045 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[5] Garay L. J., “Quantum gravity and minimum length”, Internat. J. Modern Phys. A, 10 (1995), 145–165 ; gr-qc/9403008 | DOI

[6] Gross D. J., Mende P. F., “String theory beyond the Planck scale”, Nuclear Phys. B, 303 (1988), 407–454 | DOI | MR

[7] Maggiore M., “A generalized uncertainty principle in quantum gravity”, Phys. Lett. B, 304 (1993), 65–69 ; hep-th/9301067 | DOI | MR

[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 pp., ages | DOI | MR

[9] Dadić I., Jonke L., Meljanac S., “Harmonic oscillator with minimal length uncertainty relations and ladder operators”, Phys. Rev. D, 67 (2003), 087701, 4 pp., ages ; hep-th/0210264 | DOI

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[13] Akhoury R., Yao Y.-P., “Minimal length uncertainty relation and the hydrogen spectrum”, Phys. Lett. B, 572 (2003), 37–42 ; hep-ph/0302108 | DOI | Zbl

[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 pp., ages ; hep-th/0502222 | DOI

[15] Nouicer K., “Pauli-Hamiltonian in the presence of minimal lengths”, J. Math. Phys., 47 (2006), 122102, 11 pp., ages | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 pp., ages ; quant-ph/0603077 | MR | Zbl

[19] Spector D., “Minimal length uncertainty relations and new shape invariant models”, J. Math. Phys., 49 (2008), 082101, 8 pp., ages ; arXiv:0707.1028 | DOI | MR | Zbl

[20] Bender C. M., Boettcher S., “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry”, Phys. Rev. Lett., 80 (1998), 5243–5246 ; ; Bender C. M., Boettcher S., “Quasi-exactly solvable quartic potential”, J. Phys. A: Math. Gen., 31 (1998), L273–L277 ; physics/9712001physics/9801007 | DOI | MR | Zbl | DOI | MR | Zbl

[21] Makris K. G., El-Ganainy R., Christodoulides D. N., Musslimani Z. H., “Beam dynamics in $\mathcal{PT}$ symmetric optical lattices”, Phys. Rev. Lett., 100 (2008), 103904, 4 pp., ages | DOI

[22] Klaiman S., Günther U., Moiseyev N., “Visualization of branch points in $\mathcal{PT}$ symmetric waveguides”, Phys. Rev. Lett., 101 (2008), 080402, 4 pp., ages ; arXiv:0802.2457 | DOI | MR

[23] Swanson M. S., “Transition elements for a non-Hermitian quadratic Hamiltonian”, J. Math. Phys., 45 (2004), 585–601 | DOI | MR | Zbl

[24] Cooper F., Khare A., Sukhatme U. P., Supersymmetry in quantum mechanics, World Scientific, 2002 | MR

[25] Mostafazadeh A., “Pseudo-Hermiticity versus $\mathcal{PT}$ symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian”, J. Math. Phys., 43 (2002), 205–214 ; ; Mostafazadeh A., “Pseudo-Hermiticity versus $\mathcal{PT}$-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum”, J. Math. Phys., 43 (2002), 2814–2816 ; ; Mostafazadeh A., “Pseudo-Hermiticity versus $\mathcal{PT}$-symmetry. III. Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries”, J. Math. Phys., 43 (2002), 3944–3951 ; math-ph/0107001math-ph/0110016math-ph/0203005 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[26] Kamran N., Olver P. J., “Lie algebras of differential operators and Lie-algebraic potentials”, J. Math. Anal. Appl., 145 (1990), 342–356 | DOI | MR | Zbl

[27] Quesne C., “Comment: 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], J. Phys. A: Math. Gen., 32 (1999), 6705–6710 ; ; 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 math-ph/9911004 | DOI | MR | Zbl | DOI | MR | Zbl