Repositorio Institucional del CAB-IB

Método ab-initio para el problema de tres cuerpos / An ab-initio method for the three body problem

Randazzo, Juan M. (2009) Método ab-initio para el problema de tres cuerpos / An ab-initio method for the three body problem. Tesis Doctoral en Física, Universidad Nacional de Cuyo, Instituto Balseiro.

Vista previa

PDF
Español
11Mb

Resumen en español

En este trabajo se desarrolló una metodología de expansión de las funciones de onda asociadas al problema de tres cuerpos cuántico, en el cual dos de las partículas involucradas son livianas, y la tercera pesada. Este problema es adecuado para el tratamiento de sistemas atómicos, como por ejemplo los estados ligados del átomo de He o la ionizaci ón de átomos de un electrón por impacto de electrones (e,2e). La técnica empleada para la solución de la ecuación de Schrödinger consistió en la aplicación del método de Configuración Interacción (CI ), en el cual la función de onda total se escribe como una superposición de configuraciones electrónicas separables, expresadas en coordenadas esféricas. Las funciones angulares empleadas fueron los armónicos biesféricos, autofunciones de los operadores del cuadrado del momento angular total y de la proyección del mismo a lo largo de un eje fijo en el espacio. Como base radial se utilizó el conjunto de autofunciones de un problema Sturmiano de dos cuerpos. En la ecuación que satisfacen las funciones Sturmianas la energía se fijó externamente, mientras que como autovalor se consideró a un parámetro multiplicativo de un potencial de corto alcance. También se agregó a dicha ecuación un potencial de largo alcance, que permitió incorporar al comportamiento asintótico de la base la influencia Coulombiana deseada. Empleando distintas estrategias numéricas, pudo imponerse a la base condiciones asintóticas generales, que resultaron independientes del autovalor. También se incluyeron en la ecuación potenciales arbitrarios, que permitieron generalizar la interacción entre las partículas livianas y la pesada. Presentamos el método de Configuración Interacción con funciones Sturmianas (CIFS), para el cálculo de energías y funciones de onda de los estados ligados del átomo de helio e ion del hidrógeno. Se mostró la versatilidad del método en su aplicaci ón a sistemas de dos electrones ligados para dos modelos de confiinamiento. Por último desarrollamos una metodología de expansión de la función de scattering para el problema de fragmentación en sistemas de tres cuerpos. Para ésto se utilizó una base de funciones Sturmianas con condiciones de flujo no nulo. Se estudió la convergencia del método mediante su aplicación a tres modelos de fragmentación.

Resumen en inglés

In this work we present a new methodology to solve the Schrödinger equation for the three body problem, where one of the particles involved is heavy while the other two particles are light. The method is based on the Cofiguration Interaction approach (CI ), where the wave functions are expanded by means of two{body states, in spherical coordinates. The angular functions employed are the Bispherical Harmonics, which are eigenfunctions of the total angular momentum. The radial basis are solutions of a Sturmian eigenvalue problem, obtained from a two body radial equation, where the energy is externally fixed and the eigenvalue is a multiplicative factor of a short-range generating potential. That equation also includes an auxiliary potential, which can have Coulomb behaviour, and allows us to incorporate the correct asymptotic condition of the Sturmian basis set. By means of dierent numerical schemes, general asymptotic boundary conditions can be selected, which are independent of the eigenvalues. The auxiliary potential can include the interaction between the light and heavy particles, in order to remove them from the three body Schrödinger equation. Convergence of this Cofiguration Interaction method with Sturmians Functions (CISF) is tested with the expansion of the Helium and Hydrogen ion bound state eigenfunctions. Besides, we apply the method to two electron model systems with connement, to show the versatility of the method.We also employed the CISF method to compute scattering wave functions for dierent break-up processes, using Sturmians functions with outgoing and incoming asymptotic fluxes conditions.

Tipo de objeto:	Tesis (Tesis Doctoral en Física)
Palabras Clave:	Three body problem; Problema de los tres cuerpos; Particles; Partículas; Ab-initio methods; Método ab-initio
Referencias:	[1] Bray, I., Physical Review Letters 89 (2002) 273201. [2] Bray, I., Physical Review Letters 78 (1997) 4721. [3] Baertschy, M., Rescigno, T. N., Isaacs, W. A., Li, X., and McCurdy, C. W., Physical Review A 63 (2001) 22712. [4] McCurdy, C. W. and Rescigno, T. N., APS Meeting Abstracts (2001) 5039. [5] Dodd, L. R. and Greider, K. R., Physical Review 146 (1966) 675. [6] Pindzola, M. S. et al., Journal of Physics B 40 (2007) R39. [7] Mitnik, D. M., Griffin, D. C., and Pindzola, M. S., Physical Review Letters 88 (2002) 173004. [8] Whelan, C. T., Science 286 (1999) 2457. [9] Bapat, B. et al., Journal of Physics B 32 (1999) 1859. [10] Whelan, C. T., Dreizler, R. M., Macek, J. H., and Walters, H. R. J., New directions in atomic physics, Physics of atoms and molecules, Kluwer Academic/ Plenum Publishers, (New York), 1999. [11] Rotenberg, M., Annals of Physics 19 (1962) 262. [12] P G Burke, D. F. G. and Geltman, S., Journal of Physics B 2 (1969) 1142. [13] Frapiccini, A. L., Gonzalez, V. Y., Randazzo, J. M., Colavecchia, F. D., and Gasaneo, G., International Journal of Quantum Chemistry 107 (2007) 832. [14] Broad, J. T., Physical Review A 18 (1978) 1012. [15] Heller, E. J., Rescigno, T. N., and Reinhardt, W. P., Physical Review A 8 (1973) 2946. [16] Yamani, H. A. and Reinhardt, W. P., Physical Review A 11 (1975) 1144. [17] Broad, J. T., Physical Review A 31 (1985) 1494. [18] Dube, L. J. and Broad, J. T., Journal of Physics B 22 (1989) L503. [19] Heller, E. J. and Yamani, H. A., Physical Review A 9 (1974) 1201. [20] Broad, J. T., Physical Review A 26 (1982) 3078. [21] Michels, A., Boer, J. D., and Bijl, A., Physica 4 (1937) 981. [22] Laughlin, C., Journal of Physics B 37 (2004) 4085. [23] Aguilera-Navarro, V. C., Koo, E. L., and Zimerman, A. H., Revista Brasileira de Fsica 10 (1979) 2. [24] Bray, I., Bartschat, K., and Stelbovics, A. T., Physical Review A 67 (2003) 060704. [25] Aunola, M., Journal of Physics A 38 (2005). [26] Avery, J. and Avery, J., Generalized Strumians and Atomic Spectra, Word Scientific, Denmark, 2006. [27] Manakov, N., Ovsiannikov, V., and Rapoport, L., Physics Reports 141 (1986) 320 . [28] Rawitscher, G., Physical Review C 25 (1982) 2196. [29] Weinberg, S., Physical Review 131 (1963) 440. [30] Weinberg, S., Physical Review 130 (1963) 776. [31] Ovchinnikov, S. Y. and Macek, J. H., Physical Review A 55 (1997) 3605. [32] Landau, L. D. and Lifshitz, E. M., Mecanica cuantica no-relativista, Reverte, 1969, 1969. [33] Newton, R. G., Scattering Theory of Waves and Particles: Second Edition, Dover Publications, second edition, 2002. [34] Brau, F., Journal of Physics A 37 (2004) 6687. [35] Brau, F., Journal of Physics A 36 (2003) 9907. [36] Tricomi con uent hypergeometric function: Series representations (formula 07.33.06.0008), http://functions.wolfram.com/07.33.06.0008.01. [37] van Haeringen, H., Physical Review A 18 (1978) 56. [38] Generalized laguerre function (07.03.02.0001.01), http://functions.wolfram.com/07.03.02.0001.01. [39] Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 9th Dover printing, tenth GPO printing edition, 1964. [40] Macek, J. H. and Ovchinnikov, S. Y., Physical Review A 54 (1996) 544. [41] Gasaneo, G. and Colavecchia, F. D., Journal of Physics A 36 (2003) 8443. [42] Lapack { linear algebra package, http://www.netlib.org/lapack/. [43] van Haeringen, H. and van Wageningen, R., Journal of Mathematical Physics 16 (1975) 1441. [44] van Haeringen, H., Journal of Mathematical Physics 17 (1976) 995. [45] Luk, F. T. and Qiao, S., Linear Algebra and its Applications 316 (2000) 171. [46] Rescigno, T. N., Baertschy, M., Isaacs, W. A., and McCurdy, C. W., Science 286 (1999) 2474. [47] Collins, S. A. et al., Physical Review A 65 (2002) 052717. [48] Domke, M. et al., Physical Review Letters 66 (1991) 1306. [49] Rösel, T. et al., Journal of Physics B 24 (1991) 3059. [50] Röder, J. et al., Physical Review A 53 (1996) 225. [51] Rodriguez, K. V., Gasaneo, G., and Mitnik, D. M., J. Phys B: At. Mol. Opt. Phys. 40 (2007) 3923. [52] Bromley, M. W. J. and Mitroy, J., Int. J. Quant. Chem 107 (2007) 1150. [53] Mitnik, D. M., Physical Review A 70 (2004) 022703. [54] Forrey, R. C., Physical Review A 69 (2004) 022504. [55] Frolov, A. M., Journal of Physics A: Mathematical and Theoretical 40 (2007) 6175. [56] Foumouo, E., Kamta, G. L., Edah, G., and Piraux, B., Physical Review A 74 (2006) 063409. [57] Avery, J. and Avery, J., J. Math. Chem 33 (2003) 145. [58] Baertschy, M., Rescigno, T. N., and McCurdy, C. W., Physical Review A 64 (2001) 22709. [59] Rescigno, T. N., McCurdy, C. W., Isaacs, W. A., and Baertschy, M., Physical Review A 60 (1999) 3740. [60] McCurdy, C. W. and Rescigno, T. N., Physical Review A 62 (2000) 32712. [61] Randazzo, J. M., Frapiccini, A. L., Colavecchia, F. D., and Gasaneo, G., Physical Review A 79 (2009) 022507. [62] Kheifets, A. S. and Bray, I., Physical Review A 57 (1998) 2590. [63] Kheifets, A. S. and Bray, I., Physical Review A 54 (1996) R995. [64] Avery, J. and Avery, J., J. Phys. Chem. A 108 (2004) 8848. [65] Avery, J. and Avery, J., J. Math. Chem 33 (2003) 145. [66] Avery, J. and Shim, R., Int. J. of Q. Chem 83 (2001) 1. [67] Shull, H. and Lowdin, P. O., J. Chem. Phys. 30 (1959) 617. [68] Li, T. and Shakeshaft, R., Physical Review A 71 (2005) 052505. 194 Referencias [69] Yang, B., Pont, M., Shakeshaft, R., van Duijn, E., and Piraux, B., Physical Review A 56 (1997) 4946. [70] Hylleraas, E. A., Z. Phys 54 (1929) 347. [71] Gasaneo, G. and Ancarani, L. U., Phys. Rev A 77 (2008) 012705. [72] Frolov, A. M., Journal of Physics A 39 (2006) 15421. [73] Aquilanti, V. A., Chem. Phys. Lett. 1 (1997) 214. [74] Szmytkowski, R., Journal of Physics A 33 (2000) 4553. [75] Randazzo, J. M., Frapiccini, A. L., Colavecchia, F. D., and Gasaneo, G., International Journal of Quantum Chemistry 109 (2009) 125. [76] Jaskólski, W., Physics Reports 271 (1996) 1 . [77] Banerjee, A., Kamal, C., and Chowdhury, A., Physics Letters A 350 (2006) 121. [78] Seldam, C. T. and Groot, S. D., Physica 18 (1952) 891 . [79] Ludena, E. V. and Gregori, M., The Journal of Chemical Physics 71 (1979) 2235. [80] Joslin, C. and Goldman, S., Journal of Physics B. 25 (1992) 1965 . [81] FitzGerald, S. A. et al., Physical Review B 60 (1999) 6439. [82] Komatsu, K., Murata, M., and Murata, Y., Science 307 (2005) 238. [83] Ulivi, L. et al., Physical Review B (Condensed Matter and Materials Physics) 76 (2007) 161401. [84] Carravetta, M. et al., Physical Chemistry Chemical Physics 9 (2007) 4879. [85] Cohen-Tannoudji, C., Quantum mechanics, volume 2, Wiley, 1977. [86] Clebsch{gordan coecients, http://functions.wolfram.com/HypergeometricFunctions/ClebschGordan/. [87] Avery, J., Hyperspherical Harmonics and Generalized Sturmians, Kluwer, Dordrecht, the Netherlands, 2000. [88] Temkin, A., Physical Review 126 (1962) 130. [89] Temkin, A. and Hahn, Y., Physical Review A 9 (1974) 708. [90] Poet, R., Journal of Physics B 11 (1978) 3081. [91] Goldman, S. P., Physical Review Letters 78 (1996) 2325. [92] Goldman, S. P., Physical Review A (Atomic, Molecular, and Optical Physics). 55 (1998) R677. [93] Freund, D. E., Huxtable, B. D., and Morgan, J. D., Physical Review A 29 (1984) 8443. [94] Aquino, N., Flores-Riveros, A., and Rivas-Silva, J. F., Physics Letters A 307 (2003) 326 . [95] Xu, Y. B., Tan, M. Q., and Becker, U., Physical Review Letters 76 (1996) 3538. [96] Castellani, N., private communication, 2006. [97] Mitnik, D. M., Randazzo, J., and Gasaneo, G., Physical Review A 78 (2008) 062501. [98] Merzbacher, E., Quantum Mechanics, 2nd Edition, page 68, John Wiley & Sons, 2 edition, 1969. [99] Mukhamedzhanov, A. M., Kadyrov, A. S., and Pirlepesov, F., Physical Review A 73 (2006) 012713. [100] Kadyrov, A. S., Mukhamedzhanov, A. M., and Stelbovics, A. T., Physical Review A 67 (2003) 024702. [101] Massey, H. S. W. and Mohr, C. B. O., Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (1905-1934) 136 (1932) 289. [102] Kadyrov, A. S., Mukhamedzhanov, A. M., Stelbovics, A. T., and Bray, I., Physical Review A 70 (2004) 062703. [103] Rudge, M. R. H., Rev. Mod. Phys. 40 (1968) 564. [104] Kadyrov, A. S., Mukhamedzhanov, A. M., Stelbovics, A. T., Bray, I., and Pirlepesov, F., Physical Review A 68 (2003) 022703. [105] Bethe, H., Annalen der Physik 397 (1930) 325. [106] Massey, H. S. W. and Mohr, C. B. O., Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (1905-1934) 140 (1933) 613. [107] Gayet, R., Journal of Physics B 5 (1972) 483. [108] Crothers, D. S. F. and McCann, J. F., Journal of Physics B 16 (1983) 3229. [109] Garibotti, C. R. and Miraglia, J. E., Physical Review A 21 (1980) 572. [110] M Brauner, J. S. B. and Klar, H., Journal of Physics B 22 (1989) 2265. [111] Alt, E. O. and Mukhamedzhanov, A. M., Physical Review A 47 (1993) 2004. [112] Bartschat, K. and Bray, I., Journal of Physics B 29 (1996) L577. [113] Burke, P. G., Noble, C. J., and Scott, P., Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934-1990) 410 (1987) 289. [114] Alhaidari, A. D., Heller, E. J., Yamani, H. A., and Abdelmonem, M. S., The J-matrix Method: Developments and Applications, Springer, 1 st edition, 2008. [115] S Watanabe, Y. H. and Kato, D., Journal of Physics B; Atomic, Molecular and Optical Physics 26 (1993) L495. [116] Bray, I., Physical Review A 60 (1999) 5118. [117] Burke, P. G. and Mitchell, J. F. B., Journal of Physics B 6 (1973) 320. [118] Oza, D. H. and Callaway, J., Physical Review A 27 (1983) 2840. [119] Bray, I., Konovalov, D. A., and McCarthy, I. E., Physical Review A 43 (1991) 5878. [120] Bray, I. and Stelbovics, A. T., Physical Review Letters 69 (1992) 53. [121] Bray, I. and Stelbovics, A. T., Physical Review Letters 70 (1993) 746. [122] D A Konovalov, I. B. and McCarthy, I. E., Journal of Physics B 27 (1994) L413. [123] Callaway, J. and Oza, D. H., Physical Review A 29 (1984) 2416. [124] Nuttall, J. and Cohen, H. L., Physical Review 188 (1969) 1542. [125] Rescigno, T. N. and McKoy, V., Physical Review A 12 (1975) 522. [126] McCurdy, C. W., Physical Review A 21 (1980) 464. [127] McCurdy, C. W., Stroud, C. K., and Wisinski, M. K., Physical Review A 43 (1991) 5980. [128] Rescigno, T. N., Baertschy, M., Byrum, D., and McCurdy, C. W., Physical Review A 55 (1997) 4253. [129] McCurdy, C. W., Rescigno, T. N., and Byrum, D., Physical Review A 56 (1997) 1958. [130] McCurdy, C. W. and Rescigno, T. N., Physical Review A 56 (1997) R4369. [131] Baertschy, M., Rescigno, T. N., Isaacs, W. A., and McCurdy, C. W., Physical Review A 60 (1999) R13. [132] Bartlett, P. L., Complete numerical solution of electron-hydrogen collisions, PhD thesis, Murdoch University, Perth, Western Australia, 2005., 2005. [133] C W McCurdy, M. B. and Rescigno, T. N., Journal of Physics B 37 (2004) R137. [134] McCurdy, C. W. and Rescigno, T. N., Physical Review A 62 (2000) 032712. [135] Bartlett, P. L. et al., Physical Review A 68 (2003) 020702. [136] Bottcher, C., Journal of Physics B 14 (1981) L349. [137] Wannier, G. H., Physical Review 90 (1953) 817. [138] Bottcher, C., Journal of Physics B 15 (1982) L463. [139] Pindzola, M. S. and Robicheaux, F., Physical Review A 54 (1996) 2142. 198 Referencias [140] Pindzola, M. S. and Schultz, D. R., Physical Review A 53 (1996) 1525. [141] Ihra, W., Draeger, M., Handke, G., and Friedrich, H., Physical Review A 52 (1995) 3752. [142] Peterkop, R. and Rabik, L., Journal of Physics B 5 (1972) 1823. [143] Morse, P. M. and Feshbach, H., Methods of theoretical physics, McGraw-Hill, NY, 1953. [144] Hankel function of the rst kind { from wolfram mathworld, http://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html. [145] Bessel function of the rst kind { from wolfram mathworld, http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html. [146] Bessel function of the second kind { from wolfram mathworld, http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html. [147] Rescigno, T. N. and McCurdy, C. W., Physical Review A 62 (2000) 032706. [148] Pindzola, M. S. and Robicheaux, F., Physical Review A 55 (1997) 4617. [149] Pindzola, M. S., Mitnik, D., and Robicheaux, F., Physical Review A 62 (2000) 062718. [150] von Neumann, J. and Wigner, E., Physikalische Zeitschrift 30 (1929) 465, Reprinted in [157, Paper 23]. [151] Lanczos, C., J. Res. Natl. Bur. Stand. 45 (1950) 255. [152] Komzsik, L., The Lanczos Method: Evolution and Application, Society for Industrial Mathematics, illustrated edition edition, 1987. [153] Sorensen, D. C. and Sorensen, D. C., IN 2 (1995) 23. [154] Cullum, J. K. and Willoughby, R. A., SIAM Journal on Matrix Analysis and Applications 17 (1996) 83. [155] Vandevoorde, D., A fast exponential decomposition algorithm and its applications to structured matrices, PhD thesis, Rensselaer Polytechnic Institute, Troy, NY, USA, 1996. [156] V. Hernández J. E. Roman, A. T. and Vidal, V., Single vector iteration methods in slepc (scalable library for eigenvalue problem computations), Technical report, Universidad Politécnica de Valencia, 2005. [157] Taub, A. H., editor, John von Neumann: Collected Works: Volume I: Logic, Theory of Sets and Quantum Mechanics, pub-PERGAMON, pub- PERGAMON:adr, 1961.
Materias:	Física Física > Partículas
Divisiones:	Investigación y aplicaciones no nucleares > Física > Colisiones atómicas
Código ID:	265
Depositado Por:	Marisa G. Velazco Aldao
Depositado En:	13 May 2011 10:06
Última Modificación:	13 May 2011 10:07

Personal del repositorio solamente: página de control del documento