, has been calculated in 2 and 3 dimensions. It has been ascertained that it does not exist in one dimension or, at least, that it is not computable. In this work, after giving simple arguments on the existence of this critical moment, we compute 
in one dimension.]]>
A simple inquiry on the critical electric dipole moment in one space dimension
B. Jaramilloa, H.N. Núñez–Yépezb, A.L. Salas–Britoc
a Instituto de Física, Universidad de Guanajuato, Loma del Bosque 103, Fracc. Lomas del Campestre, 37150 León, Guanajuato, e–mail: benraz@licifug.ugto.mx
b Departamento de Física, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Apartado Postal 55–534, 09340, Iztapalapa D.F. México, e–mail: nyhn@xanum.uam.mx
c Laboratorio de Sistemas Dinámicos, Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21–267, 04000, Coyoacán D.F. México, e–mail: asb@correo.azc.uam.mx
]]> Recibido el 4 de enero de 2010
Abstract
The magnitude of an electric dipole moment must be larger or equal to a certain critical value to support bound states. This is not a widely known fact that nevertheless is easy to understand on heuristic terms and relatively easy to calculate. This critical dipole moment, 
, has been calculated in 2 and 3 dimensions. It has been ascertained that it does not exist in one dimension or, at least, that it is not computable. In this work, after giving simple arguments on the existence of this critical moment, we compute in one dimension.
Keywords: 1D critical electric dipole; 1D quantum system.
Resumen
El valor de un momento dipolar eléctrico debe ser mayor o igual a un valor crítico para que admita estados ligados. Este no muy conocido hecho puede comprenderse en forma muy simple y su valor calculado en forma relativamente simple como lo hacemos en este trabajo. Se ha calculado el momento crítico en 2 y 3 dimensiones y se ha sugerido que no existe en una dimensión o que, al menos, no se le puede calcular. Damos argumentos simples para argüir su existencia y lo calculamos exactamente en una dimensión.
Descriptores: Dipolo eléctrico crítico en una dimensión; sistemas cuánticos unidimensionales.
]]>PACS: 03.65.Ge; 11.30.–j
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Acknowledgements
B.J. acknowledges the support of the Academia Mexicana de Ciencias through the Verano de la Ciencia program. This work was partially supported by CONACyT (grants 8521 and 24229) and a PAPIIT–UNAM research grant (IN 115406–3). We acknowledge with thanks the comments of Roberto Sussman. Last but not least, we dedicate this work to the memories of our good friends J. Vitela–Escamilla and G.R. Maya.
References
1. E. Fermi and E. Teller, Phys.Rev. 72 (1947) 399. [ Links ] These authors realize that an electric dipole supports bound states but only if the dipole moment exceeds a critical value. They offer no proof, only state the result. The story of the calculation and of Fermi involvement on the problem can be found in Ref. 2.
2. J.E. Turner, Am.J.Phys. 45 (1977) 758. [ Links ]
3. A.M. Essin and D.J. Griffiths, Am.J.Phys. 74 (2006) 109. [ Links ]
4. M. Avila–Aoki, C. Cisneros, R.P. Martínez–y–Romero, H.N. Núñez–Yépez, and A.L. Salas–Brito, Phys. Lett. A 373 (2009) 418. [ Links ]
5. K. Connolly and D.J. Griffiths, Am.J.Phys. 75 (2007) 524. and the references therein. [ Links ]
6. B. Jaramillo, H.N. Núñez–Yépez, and A.L. Salas–Brito, Phys. Lett. A 374 (2010) 27072708. [ Links ]
7. S. Schippers etal., J.Phys.B:At.Mol.Opt.Phys. 36 (2006) 3371. [ Links ]
8. V.P. Lide (Editor), CRC Hanbook of Physics and Chemistry, (CRC Press, 2005, Boca Ratón USA). [ Links ]
9. W.H. Mittleman and V.P. Myerscough, Phys.Lett. 23 (1966) 545. [ Links ]
10. J.E. Turner and K. Fox, Phys.Lett. 23 (1966) 547. [ Links ]
11. J.–M. Levy–Leblond, Phys.Rev. 153 (1967) 1. [ Links ]
12. B. Jaramillo, R.P. Martínez–y–Romero, H.N. Núñez–Yépez, and A.L. Salas–Brito, Phys. Lett. A 374 (2009) 150. [ Links ]
13. H.N. Núñez–Yépez, C.A. Vargas, and A.L. Salas–Brito, Eur.J. Phys. 8 (1987) 189. [ Links ]
14. W. Fischer, H. Lescke, and P. Müller, J.Math.Phys. 36 (1995) 2313 [ Links ]
15. H.N. Núñez–Yépez, and A.L. Salas–Brito, Eur.J.Phys. 8 (1987) 307. [ Links ]
16. I. Tsutsui, T. Fülöp, and Cheon Taksu, J.Phys.A:Math.Gen. 36 (2003) 275. [ Links ]
17. P. Garbaczewski, and W. Karwowski, Am.J.Phys. 72 (2004) 924. [ Links ]
18. C. Cisneros, R.P. Martínez–y–Romero, H.N. Núñez–Yépez, and A.L. Salas–Brito, Eur. J. Phys. 19 (1998) 237. [ Links ]
19. L. Kong Wan, From micro to macroquantum systems (Imperial College, 2006, London UK). [ Links ]
20. This calculation is done for the radial part of the three dimensional point dipole in Ref. 5. For the the one–dimensional –a/x2 potential see Refs. 3 and 4.
]]>21. H.E. Camblong, C.R. Ordoñez, H. Fanchiotti, and C. García–Canal, Phys. Rev. Lett. 87 (2001) 220402. [ Links ]
22. E. Turner, K. Fox, On the Bound States of an Electronin the Field of a Permanent Electric Dipole ORNL–3895 (Oak Ridge National Laboratory, 1965, Oak Ridge USA) [ Links ]
23. V.S. Vrkljan, Z.Phys. 52 (1928) 735. [ Links ]
24. V.S. Vrkljan, Z.Phys. 54 (1929) 133. [ Links ]
25. R.P. Martínez–y–Romero, H.N. Núñez–Yépez, and A.L. Salas–Brito, Rev. Mex. Fis. 35 (1989) 617. [ Links ]
26. L.W. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge UK, 1996). [ Links ]
27. P.M. Platzman, and M.I. Dykman, Science 284 (1999) 1967. [ Links ]
28. M.I. Dykman, P.M. Platzman, and P. Seddighrad, Phys.Rev.B 67 (2003) 155402, and references therein. [ Links ]
29. M.M. Nieto, Phys.Rev.A 61 (2000) 034901. [ Links ]
30. C. Störmer, Arch.Sci.Phys.Nat. 24 (1907) 113. [ Links ]
31. C. Störmer, Thepolaraurora (Clarendon Press, 1955, Oxford UK). [ Links ]
32. L. Jimenez–Lara and E. Piña, Celestial Mech. 49 (1990) 327. [ Links ]
33. A.N. Krylov, Phys.Uspekhi 36 (1993) 226. [ Links ] ]]>