Abstract
We show that the energy spectrum of the one-dimensional Dirac equation, in the presence of an attractive vectorial delta potential, exhibits a resonant behavior when one includes an asymptotically spatially vanishing weak electric field associated with a hyperbolic tangent potential. We solve the Dirac equation in terms of Gauss hyper-geometric functions and show explicitly how the resonant behavior depends on the strength of the electric field evaluated at the support of the point interaction. We derive an approximate expression for the value of the resonances and compare the results calculated for the hyperbolic potential with those obtained for a linear perturbative potential. Finally, we characterize the resonances with the help of the phase shift and the Wigner delay time.
Similar content being viewed by others
References
W. Greiner, B. Müller, J. Rafelski, Quantum Electrodynamics of Strong fields (Springer, Berlin, 1985)
J. Rafelski, P. Fulcher, A. Klein, Phys. Rep. 38, 227 (1978)
A.A. Grib, S.G. Mamaev, V.M. Mostepanenko, Vacuum Quantum Effects in Strong Fields (Friedmann Lab, St. Petersburg, 1994)
O. Klein, Z. Phys. 53, 157 (1929)
F. Sauter, Z. Phys. 69, 742 (1931)
W. Heisenberg, H. Euler, Z. Phys. 98, 718 (1936)
W. Pieper, W. Greiner, Z. Phys. 218, 327 (1969)
S.S. Gershtein, Ya.B. Zeldovich, Sov. Phys. JETP 30, 358 (1970)
W. Greiner, J. Reinhardt, Phys. Scr. T 56, 203 (1995)
V.M. Villalba, L.A. González-Díaz, Phys. Scr. 75, 645 (2007)
F. Cannata, A. Ventura, Phys. Lett. A. 372, 941 (2008)
Y. Jiang, S.-H. Dong, A. Antillón, M. Lozada-Cassou, Eur. Phys. J.C. 45, 525 (2005)
B. Sutherland, D.C. Mattis, Phys. Rev. A 24, 1194 (1981)
B.H.J. McKellar, G.J. Stephenson, Phys. Rev. A. 36, 2566 (1987)
B.H.J. McKellar, G.J. Stephenson, Phys. Rev. C. 35, 2262 (1987)
F. Domínguez-Adame, E. Maciá, J. Phys. A: Math. Gen. 22, L419 (1989)
M.G. Calkin, D. Kiang, Y. Nogami, Am. J. Phys. 55, 737 (1987)
Y. Nogami, N. Parent, F.M. Toyama, J. Phys. A: Math. Gen. 23, 56667 (1990)
A. Galindo, P. Pascual, Quantum Mechanics, vol. II (Springer, Heidelberg, 1990)
E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Part II (Oxford University Press, Oxford, 1958)
W.E. Brittin, Lectures in Theoretical Physics, vol. IV (Interscience, New York, 1962), p. 460
W. Greiner, Relativistic Quantum Mechanics, Wave Equations (Springer, New York, 1990)
W. Elberfeld, M. Kleber, J.Z. Phys. B: Cond. Matter 73, 23 (1988)
R.M. Cavalcanti, P. Giacconi, R. Soldati, J. Phys. A 36, 12065 (2003)
G.V. Dunne, C.S. Gauthier, Phys. Rev. Phys. Rev. A 69, 053409 (2004)
M. Abramowitz, I. Stegun, Handbook of Integrals, Series and Products (Dover, New York, 1965)
A.J.F. Siegert, Phys. Rev. 56, 750 (1939)
R.G. Newton, Scattering Theory of Waves and Particles (Dover, New York, 2002)
M. Goldberger, K. Watson, Collision Theory (Dover, New York, 2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Villalba, V.M., González-Díaz, L.A. Particle resonance in the Dirac equation in the presence of a delta interaction and a perturbative hyperbolic potential. Eur. Phys. J. C 61, 519–525 (2009). https://doi.org/10.1140/epjc/s10052-009-0999-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjc/s10052-009-0999-x