Darboux–deformed barriers and resonances in quantum mechanics

N. Fernández–García

Physics Department, Cinvestav, Apartado Postal 14–740 07000 México D.F., México, e–mail: jnicolas@fis.cinvestav.mx

Recibido el 1 de mayo de 2006
Aceptado el 1 de noviembre de 2006

Abstract

Scattering states in the continuum are used as Darboux transformation functions to deform square barrier potentials. The results include complex as well as real new potentials. It is shown that an appropriate superposition of Breit–Wigner distributions connects the transmission coefficient of one dimensional short range potentials.

Keywords: Darboux transformations; Gamow vectors; Breit–Wigner ditribution.

Resumen

Se usan estados de dispersión como funciones de transformación en el método de Darboux para obtener nuevos potenciales (reales o complejos) a partir de barreras cuadradas. Se muestra que una superposición apropiada de distribuciones de Breit–Wigner permite construir una buena aproximación del coeficiente de transmisión.

Descriptores: Transformaciones de Darboux; vectores de Gamow; distribución de Breit–Wigner.

PACS: 03.65.Ca; 03.65.Ge; 03.65.Fd

DESCARGAR ARTÍCULO EN FORMATO PDF

Acknowledgements

The author is indebted to Prof. O Rosas–Ortiz for critical observations and remarks. The support of CONACyT projects 50766, 49253–F, and Cinvestav is acknowledged.

References

1. G. Gamow, Z Phys. 51 (1928) 204.        [ Links ]

2. E.U. Condon and R.W. Gurney, Phys. Rev. 33 (1928) 127.        [ Links ]

3. A.J.F. Siegert, Phys. Rev. 56 (1939) 750.        [ Links ]

4. R.M. Cavalcanti and C.A.A. Carvalho, quant–ph/9608039; On the effectiveness of Gamow's method for calculating decay rates, Rev. Bras. de Fis. 21 (1999) 464.        [ Links ]

5. E.C.G. Sudarshan, Phys. Rev. A 46 (1992) 37.        [ Links ]

6. G. Breit and E.P. Wigner, Phys. Rev. 49 (1936) 519.        [ Links ]

7. W. Heitler and N. Hu, Nature 159 (1947) 776.        [ Links ]

8. R.G. Newton, J Math Phys 2 (1961) 188.        [ Links ]

9. A. Bohm, M. Gadella, and G.B. Mainland, Am. J. Phys. 57 (1989) 1103;         [ Links ] A. Bohm and M. Gadella, Dirac kets, Gamow Vectors and Gelfand Triplets, in: Lecture Notes in Physics vol 348 (New York: Springer 1989);         [ Links ] M. Gadella and R. de la Madrid, Am. .J. Phys. 70 (2002) 626.        [ Links ]

10. P. Dirac, The principles of Quantum Mechanics (London: Oxford University Press 1958);         [ Links ] K. Maurin, Generalized eigen–function expansions and unitary representations of topological groups (Warsav: Pan Stuvowe Wydawn 1968);         [ Links ] I.M. Gelfand and N. Y Vilenkin, Generalized Functions V4 (New York: Academic Press 1968).        [ Links ]

11. A. Ramírez and B. Mielnik, Rev. Mex. Fís. 49S2 (2003) 130.        [ Links ]

12. Y.B. Zel'dovich, Soviet Physics JETP 12 (1961) 542.        [ Links ]

13. N. Hokkyo, Prog. Theor. Phys 33 (1965) 1116.        [ Links ]

14. T. Berggren, Nucl. Phys A 109 (1968) 265.        [ Links ]

15. W.J. Romo, Nucl. Phys A 160 (1968) 618.        [ Links ]

16. I. Arafeva et al. (Eds), Special Issue dedicated to the subject of the International Conference on Progress in Supersymmetric Quantum Mechanics PSQM'03,         [ Links ]J. Phys A: Math. Gen. 37 (2004).        [ Links ]

17. D.J. Fernández, R. Muñoz, and A. Ramos, Phys. Lett. A 308 (2003) 11.        [ Links ]

18. R. Muñoz, Complex Super symmetric Transformations and Exactly Solvable Potentials (Mexico: PhD dissertation, Physics Department, Cinvestav 2004).        [ Links ]

19. O. Rosas–Ortiz and R. Muñoz, J. Phys. A: Math. Gen. 36 (2003) 8497.        [ Links ]

20. O. Rosas–Ortiz, Rev. Mex. Fis. (at press)        [ Links ]

21. R. Muñoz, Phys. Lett. A 345 (2005) 287.        [ Links ]

22. N. Fernández–García, Transformaciones de Gamow Supersimetricas en Mecánica Cuántica (Mexico: MSC dissertation, Physics Department, Cinvestav 2005).        [ Links ]

23. N. Fernández–García and O. Rosas–Ortiz, On Gamow vectors and Darboux transformations in quantum mechanics, in preparation.        [ Links ]

24. A. Bohm, M. Gadella, and P. Kielanowski (Eds), Int. J. Theor. Phys. 42 (2003).        [ Links ]

25. M. Pierronne and L. Márquez, Z Physik A 286 (1078) 19.        [ Links ]

26. F. Cannata, G. Junker, and J. Trost, Phys Lett A 246 (1998) 219.        [ Links ]

27. A.A. Andrianov, M.V. Ioffe, F. Cannata, and J.P Dedonder, Int. J. Mod. Phys 14 (1999) 2675.        [ Links ]

28. C.V. Sukumar, Supersymmetric Quantum Mechanics and its Aplications, in: R. Bijker et al. (Eds) AIP Conference Proceedings 744 (New York 2005).        [ Links ]

29. J.R. Taylor, Scattering Theory (John Wiley and sons, New York 1972) p. 257.        [ Links ]

30. L. de la Peña, Introducción a la Mecánica Cuántica, Segunda Edición (Fondo de Cultura Económica, 1991).        [ Links ]

31. D.J. Fernández and N. Fernández–García, Higher–Order Supersymetric Quantum Mechanics, in: R. Bijker et al. (Eds), AIP Conference Proceedings 744 (New York 2005).        [ Links ]

32. B. Mielnik and O. Rosas–Ortiz, J. Phys A: Math. Gen. 37 (2004) 10007.        [ Links ]

33. J.M. Lévi–Leblond and F. Ballibar, Quantics, Rudiments of Quantum Physics (Amsterdam: North Holland 1990).        [ Links ]

34. R. de la Madrid, J. Phys A: Math. Gen. 37 (2004) 8129;         [ Links ] R. de la Madrid, Eur. J. Phys 26 (2005) 287.        [ Links ]

35. D.J. Fernández and H. Rosu, Rev. Mex. Fís. 46S2 (2000) 153;         [ Links ] D.J. Fernández and H. Rosu, Phys. Scr 64 (2001) 177.        [ Links ]