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

SIGMA 7 (2011), 044, 17 pages      arXiv:1104.5695
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time

Nicolás Fernández-García a and Oscar Rosas-Ortiz b
a) Instituto de Física, UNAM, AP 20-353, 01000 México D.F., Mexico
b) Physics Department, Cinvestav, A.P. 14-740, México DF 07000, Mexico

Received December 01, 2010, in final form April 29, 2011; Published online May 05, 2011

We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases in which the impinging particle is 'captured' by the semi-harmonic rectangular potentials. The 'time of capture' is connected with the dwell time of the scattered wave. Using the particle absorption method, it is shown that the dwell time τDa coincides with the phase time τW of Eisenbud and Wigner, calculated as the energy derivative of the reflected wave phase shift. Analytical expressions are derived for the phase time τW of the semi-harmonic delta well and barrier potentials.

Key words: exactly solvable potentials; scattering process; resonances; Eisenbud-Wigner phase time; dwell time.

pdf (586 kb)   tex (958 kb)


  1. Bastard G., Theoretical investigations of super-lattice band-structure in the evelope-function approximation, Phys. Rev. B 25 (1982), 7584-7597.
  2. Capasso F., Sirtori C., Faist J., Sivco D.L., Chu S.-N.G., Cho A.Y., Observation of an electronic bound state above a potential well, Nature 358 (1992), 565-567.
  3. Brändas E., Elander N. (Editors), Resonances. The unifying route towards the formulation of dynamical processes. Foundations and applications in nuclear, atomic and molecular physics, Lecture Notes in Physics, Vol. 325, Springer-Verlag, Berlin, 1989.
  4. Rosas-Ortiz O., Fernández-García N., Cruz y Cruz S., A primer on resonances in quantum mechanics, AIP Conf. Proc. 1077 (2008), 31-57, arXiv:0902.4061.
  5. Rotter I., A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A: Math. Theor. 42 (2009), 153001, 51 pages.
  6. García-Calderón G., Theory of resonant states: an exact analytical approach for open quantum systems, Adv. Quantum Chem. 60 (2010), 407-456.
  7. Fernández-García N., Rosas-Ortiz O., Extended WKB method, resonances and supersymmetric radial barriers, Int. J. Theor. Phys., to appear, arXiv:1103.3692.
  8. Espinosa M.G., Kielanowski P., Unstable quantum oscillator, J. Phys. Conf. Ser. 128 (2008), 012037, 7 pages.
  9. Ludviksson A., A simple model of a decaying quantum mechanical state, J. Phys. A: Math. Gen. 20 (1987), 4733-4738.
  10. Emmanouilidou A., Moiseyev N., Stark and field-born resonances of an open square well in a static external electric field, J. Chem. Phys. 122 (2005), 194101, 9 pages.
  11. Antoniou I.E., Gadella M., Hernández E., Jauregui A., Melnikov Y., Mondragón A., Pronko G.P., Gamow vectors for barrier wells, Chaos Solitons Fractals 12 (2001), 2719-2736.
  12. de la Madrid R., Gadella M., A pedestrian introduction to Gamow vectors, Amer. J. Phys. 70 (2002), 626-638, quant-ph/0201091.
  13. de la Madrid R., The rigged Hilbert space of the algebra of the one-dimensional rectangular barrier potential, J. Phys. A: Math. Gen. 37 (2004), 8129-8157, quant-ph/0407195.
  14. Zavin R., Moiseyev N., One-dimensional symmetric rectangular well: from bound to resonance via self-orthogonal virtual state, J. Phys. A: Math. Gen. 37 (2004), 4619-4628.
  15. Fernández-García N., Rosas-Ortiz O., Gamow-Siegert functions and Darboux-deformed short range potentials, Ann. Physics 323 (2008), 1397-1414, arXiv:0810.5597.
  16. Klaiman S., Moiseyev N., The absolute position of a resonance peak, J. Phys. B: At. Mol. Opt. Phys. 43 (2010), 185205, 4 pages, arXiv:1005.4756.
  17. Christiansen P.L., Arnbak H.C., Zolotaryuk A.V., Ermakov V.N., Gaididei Y.B., On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function, J. Phys. A: Math. Gen. 36 (2003), 7589-7600.
  18. Flügge S., Practical quantum mechanics, Springer-Verlag, Berlin, 1999.
  19. Fernández D., Gadella M., Nieto L.M., Supersymmetry transformations for delta potentials, SIGMA 7 (2011), 029, 14 pages, arXiv:1012.0808.
  20. Scarf F.L., New soluble energy band problem, Phys. Rev. 112 (1958), 1137-1140.
  21. Negro J., Nieto L.M., Rosas-Ortiz O., On a class of supersymmetric quantum mechanical singular potentials, in Foundations of Quantum Physics, Editors R. Blanco et al., CIEMAT/RSEF, Madrid, 2002, 259-270.
  22. Negro J., Nieto L.M., Rosas-Ortiz O., Regularized Scarf potentials: energy band structure and supersymmetry, J. Phys. A: Math. Gen. 37 (2004), 10079-10093.
  23. Mielnik B., Rosas-Ortiz O., Factorization: little or great algorithm?, J. Phys. A: Math. Gen. 37 (2004), 1007-10035.
  24. Khare A., Sukhatme U., Periodic potentials and supersymmetry, J. Phys. A: Math. Gen. 37 (2004), 10037-10056, quant-ph/0402206.
  25. Baye D., Sparenberg J.-M., Inverse scattering with supersymmetric quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10223-10249.
  26. Andrianov A.A., Cannata F., Nonlinear supersymmetry for spectral design in quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10297-10321, hep-th/0407077.
  27. Fernández D.J., Supersymmetric quantum mechanics, AIP Conf. Proc. 1287 (2010), 3-36, arXiv:0910.0192.
  28. Andrianov A.A., Borisov N.V., Ioffe M.V., Factorization method and Darboux transformation for multidimensional Hamiltonians, Theoret. and Math. Phys. 61 (1984), 1078-1088.
  29. Baye D., Levai G., Sparenberg J.-M., Phase-equivalent complex potentials, Nuclear Phys. A 599 (1996), 435-456.
  30. Andrianov A.A., Ioffe M.V., Cannata F., Dedonder J.-P., SUSY quantum mechanics with complex superpotentials and real energy spectra, Internat. J. Modern Phys. A 14 (1999), 2675-2688, quant-ph/9806019.
  31. Fernández D.J., Muñoz R., Ramos A., Second order SUSY transformations with 'complex energies', Phys. Lett. A 308 (2003), 11-16, quant-ph/0212026.
  32. Rosas-Ortiz O., Muñoz R., Non-Hermitian SUSY hydrogen-like Hamiltonians with real spectra, J. Phys. A: Math. Gen. 36 (2003), 8497-8506, quant-ph/0302190.
  33. Rosas-Ortiz O., Gamow vectors and supersymmetric quantum mechanics, Rev. Mexicana Fís. 53 (2007), suppl. 2, 103-109, arXiv:0810.2283.
  34. Taylor J.R., Scattering theory. The quantum theory of nonrelativistic collisions, Dover, New York, 2006.
  35. Bohm D., Quantum theory, Prentice-Hall, Englewood Cliffs, NJ, 1951.
  36. Wigner E.P., Lower limit for the energy derivative of the scattering phase shift, Phys. Rev. 98 (1955), 145-147.
  37. Smith F.T., Lifetime matrix in collision theory, Phys. Rev. 118 (1960), 349-356.
  38. Amrein W.O., Cibils M.B., Global and Eisenbud-Wigner time delay in scattering theory, Helv. Phys. Acta 60 (1987), 481-500.
  39. Richard S., Tiedra de Aldecoa R., Time delay is a common feature of quantum scattering theory, arXiv:1008.3433.
  40. Moshinsky M., Boundary conditions and time-dependent states, Phys. Rev. 84 (1951), 525-532.
  41. Moshinsky M., Diffraction in time, Phys. Rev. 88 (1952), 625-631.
  42. Osborn T.A., Bollé D., An extended Levinson's theorem, J. Math. Phys. 18 (1977), 432-440.
  43. Pedersen M.H., van Langen S.A., Büttiker M., Charge fluctuations in quantum point contacts and chaotic cavities in the presence of transport, Phys. Rev. B 57 (1998), 1838-1846, cond-mat/9707086.
  44. Emmanouilidou A., Reichl L.E., Scattering properties of an open quantum system, Phys. Rev. A 62 (2000), 022709, 8 pages.
  45. Amrein W.O., Jacquet Ph., Time delay for one-dimensional quantum systems with steplike potentials, Phys. Rev. A 75 (2007), 022106, 20 pages, quant-ph/0610198.
  46. Barr A.M., Reichl L.E., Quasibound states in two- and three-dimensional open quantum systems, Phys. Rev. A 81 (2010), 022707, 7 pages.
  47. Huage E.H., Støvneng J.A., Tunneling times: a critical review, Rev. Mod. Phys. 61 (1989), 917-936.
  48. de Carvalho C.A.A., Nussenzveig H.M., Time delay, Phys. Rep. 364 (2002), 83-174.
  49. Sassoli de Bianchi M., Time-delay of classical and quantum scattering process: a conceptual overview and a general definition, arXiv:1010.5329.
  50. Golub R., Felber S., Gähler R., Gutsmiedl E., A modest proposal concerning tunneling times, Phys. Lett. A 148 (1990), 27-30.
  51. Huang Y.Z., Wang C.M., Comparisons of phase times with tunnelling times based on absorption probabilities, J. Phys.: Condens. Matter 3 (1991), 5915-5919.
  52. Muga J.G., Brourard S., Sala R., Equivalence between tunnelling times based on (a) absorption probabilities, (b) the Larmor clock, and (c) scattering projectors, J. Phys.: Condens. Matter 4 (1992), L579-L584.
  53. Jaworski W., Wardlaw D.M., Time delay in tunneling: transmission and reflection time delays, Phys. Rev. A 37 (1988), 2843-2854.
  54. Negro J., Nieto L.M., Rosas-Ortiz O., Confluent hypergeometric equations and related solvable potentials in quantum mechanics, J. Math. Phys. 41 (2000), 7964-7996.
  55. Rosas-Ortiz O., Negro J. and Nieto L.M., Physical sectors of the confluent hypergeometric functions space, Rev. Mexicana Fís. 49 (2000), suppl. 1, 88-94, quant-ph/0105091.
  56. Wang Z.X., Guo D.R., Special functions, World Scientific, Singapore, 1989.

Previous article   Next article   Contents of Volume 7 (2011)