Two-dimensional delta potential wells and condensed-matter physics
M. de Llano a, A. Salazar b, and M.A. Solís b,c
a Instituto de Investigaciones en Materiales, UNAM Apartado Postal 70-360, 04510 México, D.F., México.
b Instituto de Física, UNAM, Apartado Postal 20-364 01000 México, D.F., México.
c Department of Physics, Washington University St. Louis, Missouri 63130, USA.
]]>Recibido el 24 de mayo de 2005.
Aceptado el 4 de agosto de 2005.
Abstract
It is well-known that a delta potential well in 1D has only one bound state but that in 3D it supports an infinite number of bound states with infinite binding energy for the lowest level. We show how this also holds for the less familiar 2D case, and then discuss why this makes 3D delta potential wells unphysical as models of interparticle interactions for condensed-matter, many-body systems. However, both 2D and 3D delta wells can be "regularized" to support a single bound level which in turn renders them conveniently simple single-parameter interactions, e.g., for modeling the pair-forming dynamics of quasi-2D superconductors such as the cuprates, or in 3D of other superconductors and of neutral-fermion superfluids such as ultra-cold trapped Fermi gases.
Keywords: Delta potential wells; bound states; regularization; condensed-matter physics.
Resumen
Es bien sabido que un pozo de potencial delta en 1D tiene un solo estado ligado pero que en 3D tiene un número infinito de estos estados con una energía de "amarre" infinita para el nivel más bajo. Aquí mostramos como esto también ocurre para el caso bidimensional, que es menos familiar, para luego discutir por que los pozos de potencial delta en 3D no son físicos como modelos de interacciones entre partículas para sistemas de muchos cuerpos en materia condensada. No obstante, ambos pozos delta, en 2D y 3D, pueden ser regularizados para soportar un solo nivel ligado, lo cual los convierte convenientemente en interacciones de un solo parámetro, por ejemplo, para modelar la dinámica de formación de pares en superconductores casi-bidimensionales tales como los cupratos, o en 3D la formación de pares en otros superconductores y en superfluidos fermiónicos neutros tales como los gases de Fermi atrapados ultrafríos.
]]> Descriptores: Pozo de potencial delta en 2D; regularización; superconductores; superfluidos.
PACS: 03.75.Ss; 03.65.-w; 03.65.Ge; 74.78.-w
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Acknowledgments
We thank M. Fortes and O. Rojo for discussions, and acknowledge UNAM-DGAPA-PAPIIT (Mexico) grant # IN106401, and CONACyT (Mexico) grant # 27828 E, for partial support. MAS thanks Washington university, St. Louis, MO, USA, for hospitality during a sabbatical year.
References
]]>1. G. Stan, S.M. Gatica, M. Boninsegni, S. Curtarolo, and M.W. Cole, Am. J. Phys. 67 (1999) 1170. [ Links ]
2. E. Corcoran and G. Zopette, Sci. Am. [Suppl. The Solid-State Century] (1997) 25. [ Links ]
3. L.I. Glazman, Am. J. Phys. 70 (2002) 376. [ Links ]
4. F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [ Links ]
5. P.W.Anderson, The theory of superconductivity in the High-Tc cuprates (Princeton University Press, NJ, 1997). [ Links ]
]]>6. S. Albeverio, F. Gesztesy, R. Heegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Springer-Verlag, Berlin, 1988). [ Links ]
7. R. Jackiw, in M.A.B. Bég Memorial Volume, edited by A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991). [ Links ]
8. P. Gosdzinsky and R. Tarrach, Am. J. Phys. 59 (1991) 70. [ Links ]
9. L.N. Cooper, Phys. Rev. 104 (1956) 1189. [ Links ]
10. M. Casas, M. Fortes, M. de Llano, A. Puente, and M.A. Solís, Int. J. Theor. Phys. 34 (1995) 707. [ Links ]
]]>11. S. Gasiorowicz, Quantum Physics (J. Wiley & Sons, NY, 1996) p. 93. [ Links ]
12. D.A. Park, Introduction to the Quantum Theory (McGraw-Hill, NY, 1974), p. 234. [ Links ]
13. L. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon Press, NY, 1987) p. 162. A factor of 1/2 missing in this reference is included in our Eq. (3). [ Links ]
14. For a review, see M. Randeria, in: Bose-Einstein Condensation, ed. A. Griffin et al. (Cambridge University, Cambridge, 1995) p. 355. [ Links ]
15. M.J. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87 (2001) 120406. [ Links ]
]]>16. Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89 (2002) 130402. [ Links ]
17. D.L. Goodstein, States of Matter (Dover, NY, 1985) p. 386. [ Links ]
18. E.H. Lieb and M. de Llano, J. Math. Phys. 19 (1978) 860. [ Links ]
19. V.C. Aguilera-Navarro, E. Ley-Koo, M. de Llano, S.M. Peltier, and A. Plastino, J. Math. Phys. 23 (1982) 2439. [ Links ]
20. M. Casas et al., Phys. Rev. A 44 (1991) 4915. [ Links ]
]]>21. R.M. Quick, C. Esebbag, and M. de Llano, Phys. Rev. B 47 (1993) 11512. [ Links ]
22. E. Merzbacher, Quantum Mechanics (J. Wiley & Sons, NY, 1998) p. 257. [ Links ]
23. G. Arfken, Mathematical Methods for Physicists (Academic Press, NY, 2001). [ Links ]
24. M. de Llano, Mecánica Cuántica (Facultad de Ciencias, UNAM, 2002) 2nd Edition, p. 124. [ Links ]
25. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, NY, 1964). [ Links ]
]]>26. B.S. Deaver, Jr. and W.M. Fairbank, Phys. Rev. Lett. 7 (1961) 43. [ Links ]
27. R. Doll and M. Näbauer, Phys. Rev. Lett. 7 (1961) 51. [ Links ]
28. C.E. Gough et al., Nature 326 (1987) 855. [ Links ]
29. K. Chadan, N.N. Khuri, A. Martin, and Tai Tsun Wu, J. Math. Phys. 44 (2003) 406. [ Links ]
30. K. Miyake, Prog. Teor. Phys. 69 (1983) 1794. [ Links ]
]]>31. R.M. Carter et al., Phys. Rev. B. 52 (1995) 16149. [ Links ]
32. S.K. Adhikari et al., Physica C 351 (2001) 341. [ Links ]
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