Electric Multipole Polarizabilities of Quantum Bound Systems in the Transition Matrix Formalism


A new general formalism for determining the electric multipole polarizabilities of quantum (atomic and nuclear) bound systems based on the use of the transition matrix in momentum space has been developed. As distinct from the conventional approach with the application of the spectral expansion of the total Green’s function, our approach does not require preliminary determination of the entire unperturbated spectrum; instead, it makes possible to calculate the polarizability of a few-body bound complex directly based on solving integral equations for the wave function of the ground bound state and the transition matrix at negative energy, both of them being real functions of momenta. A formula for the multipole polarizabilities of a two-body bound complex formed by a central interaction potential has been derived and studied. To test, the developed t-matrix formalism has been applied to the calculation of the dipole, quadrupole and octupole polarizabilities of the hydrogen atom.

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V. Kharchenko, "Electric Multipole Polarizabilities of Quantum Bound Systems in the Transition Matrix Formalism," Journal of Modern Physics, Vol. 4 No. 1, 2013, pp. 99-107. doi: 10.4236/jmp.2013.41016.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] V. Efimov, “Energy Levels Arising from Resonant Two-Body Forces in a Three-Body System,” Physics Letters, Vol. 33B, No. 8, 1970, pp. 563-564.
[2] H. Feshbach, “A Unified Theory of Nuclear Reactions II,” Annals of Physics, Vol. 19, No. 2, 1962, pp. 287-313.
[3] E. Braaten and H.-W. Hammer, “Universality in Few-Body Systems with Large Scattering Length,” Physics Reports, Vol. 428, No. 5-6, 2006, pp. 259-390. doi:10.1016/j.physrep.2006.03.001
[4] T. Kraemer, et al., “Evidence for Efimov Quantum States in an Ultracold Gas of Caesium Atoms,” Nature (London) Vol. 440, No. 7082, 2006, pp. 315-318. doi:10.1038/nature04626
[5] C. Chin, P. Grimm, P. Julienne and E. Tiesinga, “Feshbach Resonance in Ultracold Atoms,” Reviews of Modern Physics, Vol. 82, No. 2, 2010, pp. 1225-1286. doi:10.1103/RevModPhys.82.1225
[6] N. L. Rodning, L. D. Knutson, W. G. Lynch and M. B. Tsang, “Measurement of the Electric Polarizability of the Deuteron,” Physical Review Letters, Vol. 49, No. 13, 1982, pp. 909-912. doi:10.1103/PhysRevLett.49.909
[7] J. L. Friar, S. Fallieros, E. L. Tomusiak, D. Skopik and E. G. Fuller, “Electric Polarizability of the Deuteron,” Physical Review C, Vol. 27, No. 3, 1983, pp. 1364-1366. doi:10.1103/PhysRevC.27.1364
[8] F. Goeckner, L. O. Lamm and L. D. Knutson, “Measurement of the Electric Polarizability of 3He,” Physical Review C, Vol. 43, No. 1, 1991, pp. 66-72.
[9] G. A. Rinker, “Nuclear Polarization in Muonic Helium,” Physical Review A, Vol. 14, No. 1, 1976, pp. 18-29. doi:10.1103/PhysRevA.14.18
[10] J. L. Friar, “Nuclear Polarization Corrections in μ-4 He Atoms,” Physical Review C, Vol. 16, No. 4, 1977, pp. 1540- 1548. doi:10.1103/PhysRevC.16.1540
[11] K. Pachucki and A. M. Moro, “Nuclear Polarizability of Helium Isotopes in Atomic Transitions,” Physical Review A, Vol. 75, No. 3, 2007, pp. 25211-25214.
[12] J. L. Friar and S. Fallieros, “Deuteron Electric Polarizability,” Physical Review C, Vol. 29, No. 1, 1984, pp. 232- 239. doi:10.1103/PhysRevC.29.232
[13] V. F. Kharchenko, S. A. Shadchin and S. A. Permyakov, “Non-Perturbative Theory of the Polarization Interaction in Three-Body Systems,” Physics Letters B, Vol. 199, No. 1, 1987, pp. 1-4. doi:10.1016/0370-2693(87)91451-1
[14] V. F. Kharchenko and S. A. Shadchin, “Theory of Polarization Particle-Complex Interaction in the Three-Body Approach,” Ukrainian Journal of Physics, Vol. 42, No. 8, 1997, pp. 912-920.
[15] A. V. Kharchenko, “Effect of the Deuteron Anisotropy: Longitudinal and Transverse Components of the Electric Dipole Polarizability,” Nuclear Physics A, Vol. 617, No. 1, 1997, pp. 34-44. doi:10.1016/S0375-9474(97)00011-0
[16] J. L. Friar and G. L. Payne, “Nuclear Polarizabilities and Logarithmic Sum Rules,” Physical Review C, Vol. 55, No. 6, 1997, pp. 2764-2767. doi:10.1103/PhysRevC.55.2764
[17] V. D. Efros, W. Leidemann and G. Orlandini, “PhotodisIntegration of the Three-Nucleon Systems and Their Polarizabilities,” Physics Letters B, Vol. 408, No. 1, 1997, pp. 1-6. doi:10.1016/S0370-2693(97)00772-7
[18] J.-W. Chen, H. W. Grie?hammer, M. J. Savage and R. P. Springer, “The Polarizability of the Deuteron,” Nuclear Physics A, Vol. 644, No. 3, 1998, pp. 221-234. doi:10.1016/S0375-9474(98)80012-2
[19] D. R. Phillips, G. Rupak and M. J. Savage, “Improving the Convergence of NN Effective Field Theory,” Physics Letters, Vol. B473, No. 3-4, 2000, pp. 209-218.
[20] X. Ji and Y. Li, “Sum Rules and Spin-Dependent Polari- zabilities of the Deuteron in Effective Field Theory,” Physics Letters, Vol. B591, No. 1-2, 2004, pp. 76-80.
[21] D. Gazit, N. Barnea, S. Bacca, W. Leidemann and G. Orlandini, “Photonuclear Sum Rules and the Tetrahedral Configuration of 4He,” Physical Review C, Vol. 74, No. 6, 2006, pp. 10011-10015.
[22] I. Stetcu, S. Quaglioni, J. L. Friar, A. C. Hayes and P. Navratil, “Electric Dipole Polarizabilities of Hydrogen and Helium Isotopes,” Physical Review C, Vol. 79, No. 6, 2009, pp. 40011-40016.
[23] V. F. Kharchenko and A. V. Kharchenko, “Electric Polarization of the Lambda Hypertriton Nucleus,” Collected Physical Papers (Lviv), Vol. 7, 2008, pp. 432-443.
[24] V. F. Kharchenko and A. V. Kharchenko, “Electric Dipole Polarizabilities of the Triton and Lambda Hypertriton,” International Journal of Modern Physics E, Vol. 19, No. 2, 2010, pp. 225-242.
[25] A. Dalgarno and J. T. Lewis, “The Exact Calculation of Long-Range Forces between Atoms by Perturbation Theory,” Proceeding of the Royal Society A (London), Vol. 233, No. 1192, 1955, pp. 70-74. doi:10.1098/rspa.1955.0246
[26] L. D. Faddeev, “Scattering Theory for a Three-Particle System,” Soviet Physics JETP, Vol. 12, 1961, pp. 1014-1019.
[27] N. C. Francis and K. M. Watson, “The Elastic Scattering of Particles by Atomic Nuclei,” Physical Review, Vol. 92, No. 2, 1953, pp. 291-303. doi:10.1103/PhysRev.92.291
[28] C. J. Joachain, “Quantum Collision Theory,” North-Holland Publishing Company, Amsterdam, American Elsevier Publishing Co., Inc., New-York, 1975.
[29] O. A. Yakubovsky, “On the Integral Equations in the Theory of N Particle Scattering,” Soviet Journal of Nuclear Physics, Vol. 5, 1967, pp. 937-942.
[30] Y. Yamaguchi, “Two-Nucleon Problem when the Potential Is Nonlocal but Separable,” Physical Review, Vol. 95, No. 6, 1954, pp. 1628-1634. doi:10.1103/PhysRev.95.1628
[31] V. F. Kharchenko and S. A. Shadchin, “Three-Body Theory of the Effective Interaction between a Particle and a Two-Particle Bound System,” Institute for Theoretical Physics, ITP-93-24E, Kyiv, 1993.
[32] Y. Yamaguchi and Y. Yamaguchi, “Two-Nucleon Problem When the Potential Is Nonlocal But Separable II,” Physical Review, Vol. 95, No. 6, 1954, pp. 1635-1643. doi:10.1103/PhysRev.95.1635
[33] S. A. Shadchin and V. F. Kharchenko, “The Analytical Expression for the Two-Particle Coulomb Green’s Function with Explicitly Separated Singularities,” Journal Physics B, Vol. 16, No. 8, 1983, pp. 1319-1322. doi:10.1088/0022-3700/16/8/009
[34] V. A. Fock, “On the Theory of the Hydrogen Atom,” Zeitschrift für Physik, Vol. 98, No. 3-4, 1935, pp. 145-154. doi:10.1007/BF01336904
[35] L. Castillejo, I. C. Percival and M. J. Seaton, “On the Theory of Elastic Collisions between Electrons and Hydrogen Atoms,” Proceedings of the Royal Society A (London), Vol. 254, No. 1277, 1960, pp. 259-272. doi:10.1098/rspa.1960.0019

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