Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Wave Function of Nonsymmetrical Three Body Molecules in the First Excited States

Document Type : Research Paper

Authors

H Khajehazad
M.R Eskandari
Abstract
Eigenvalues and eigenvectors of the excited states of three body molecular systems contacting under the coulomb potential are calculated parametrically by the direct solution of Schrodinger equation without using any approximation or variation parameters. This has done by expressing the coordinates of system in Jacobi and then in hyperspherical coordinates and consequently by the expansion of the angular part of wave function in hyperspherical harmonics and the spherical part of the wave function in extended Laguerre functions. Thus, the Schrodinger equation for three body molecular system becomes a non-differential matrix equation for eigenvalues and eigenvectors (expansion coefficients). After computing the expansion coefficients (wave function) the expectation value of various parameters of the system such as separation between particles can be determined.

Highlights

 

 

  1. M. R. Eskandari, M. Mahdavi, The minimum binding energy and size of doubly muonic D3 molecule, Int. J. Mod. Phys. C 13 (2002) 265.

     

  2. M. R. Eskandari, F. Faghihi, Minimum binding energy and size of the doubly muonic T3 molecule, Int. J. Quantum Chem. 93 (2003) 377.

 

  1. M. R. Eskandari, M. Mahdavi, H. Khajehazad, Calculation of binding energy for non-symmetric muonic helium hydride ions in the hyperspherical approach, Phys. Rev. A 71, (2005) 042507-042513.

 

  1. D. V. Fedorov, A. S. Jensen, Efimov effect in coordinate space Faddeev equations, Phys. Rev. Lett. 71 (1993) 4103-4106.

 

  1. C. D. Lin, Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995) 1-83.

 

  1. R. Chattopadhyay, T. K. Das, Adiabatic approximation in atomic three-body systems, Phys. Rev. A 56 (1997) 1281-1287.

 

  1. M. R. Eskandari, M. Mahdavi, Calculation of binding energy for muonic three-body systems in the hyperspherical approach, Phys. Rev. A 68 (2003) 032511-032517.

 

  1. N. Barnea, A. Novoselsky, Construction of hyperspherical functions symmetrized with respect to the orthogonal and the symmetric groups, Ann. Phys. 256 (1997) 192-225.

 

  1. S. Watanabe, Y. Hosoda, D. Kato, Hyperspherical close-coupling method extended to the two-electron continuum region: test on the s-wave model for e-H scattering, J. Phys. B 26, L495 (1993).

 

  1.  Zhong-Qi Ma, An-Ying Dai, Quantum three-body problem, arXiv:physics/9905051.

 

  1.  Md. A. Khan, Hyperspherical three-body calculation for muonic atoms, Eur. Phys. J. D 66 (2012) 83.

 

  1.  Md. A. Khan, S. K. Dutta, T. K. Das, Computation of raynal-revai coefficients for the hyperspherical approach to a three-body system, Fizika B, 8 (1999) 469-482.

 

  1.  M. R. Eskandari, H. Khajehazad, Inter. J. Modern Physics E, 19 (2010) 419-435.

     

     J. L. Ballot and M. Fabre de la Ripelle, Application of the hyperspherical formalism to the trinucleon bound state problems, Ann. Phys. 127 (1980) 62-125.

     

     C. Deng, R. Zhang, D. Feng, Solution of atomic and molecular Schrödinger equation described by hyperspherical coordinates, Int. J. Quan. Chem. 45 (1993) 385.

     

     J. Raynal and J. Revai, Transformation coefficients in the hyperspherical approach to the three-body problem, Nuovo Cimento, 68 (1970) 612-622.

     

     J. Ackermann, Global and local properties of the S states of the dtμ molecular ion: A finite-element study, Phys. Rev. A 57 (1998) 4201-4203.

     

     C. D. Lin, Classification and supermultiplet structure of doubly excited states, Phys. Rev. A. 29 (1984) 1019-1033.

     

Keywords

Tree Body Systems
Excited States
Eigen Values
Eigen Vectors
Expected Values

  1.  

     

    1. M. R. Eskandari, M. Mahdavi, The minimum binding energy and size of doubly muonic D3 molecule, Int. J. Mod. Phys. C 13 (2002) 265.

       

    2. M. R. Eskandari, F. Faghihi, Minimum binding energy and size of the doubly muonic T3 molecule, Int. J. Quantum Chem. 93 (2003) 377.

     

    1. M. R. Eskandari, M. Mahdavi, H. Khajehazad, Calculation of binding energy for non-symmetric muonic helium hydride ions in the hyperspherical approach, Phys. Rev. A 71, (2005) 042507-042513.

     

    1. D. V. Fedorov, A. S. Jensen, Efimov effect in coordinate space Faddeev equations, Phys. Rev. Lett. 71 (1993) 4103-4106.

     

    1. C. D. Lin, Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995) 1-83.

     

    1. R. Chattopadhyay, T. K. Das, Adiabatic approximation in atomic three-body systems, Phys. Rev. A 56 (1997) 1281-1287.

     

    1. M. R. Eskandari, M. Mahdavi, Calculation of binding energy for muonic three-body systems in the hyperspherical approach, Phys. Rev. A 68 (2003) 032511-032517.

     

    1. N. Barnea, A. Novoselsky, Construction of hyperspherical functions symmetrized with respect to the orthogonal and the symmetric groups, Ann. Phys. 256 (1997) 192-225.

     

    1. S. Watanabe, Y. Hosoda, D. Kato, Hyperspherical close-coupling method extended to the two-electron continuum region: test on the s-wave model for e-H scattering, J. Phys. B 26, L495 (1993).

     

    1.  Zhong-Qi Ma, An-Ying Dai, Quantum three-body problem, arXiv:physics/9905051.

     

    1.  Md. A. Khan, Hyperspherical three-body calculation for muonic atoms, Eur. Phys. J. D 66 (2012) 83.

     

    1.  Md. A. Khan, S. K. Dutta, T. K. Das, Computation of raynal-revai coefficients for the hyperspherical approach to a three-body system, Fizika B, 8 (1999) 469-482.

     

    1.  M. R. Eskandari, H. Khajehazad, Inter. J. Modern Physics E, 19 (2010) 419-435.

       

       J. L. Ballot and M. Fabre de la Ripelle, Application of the hyperspherical formalism to the trinucleon bound state problems, Ann. Phys. 127 (1980) 62-125.

       

       C. Deng, R. Zhang, D. Feng, Solution of atomic and molecular Schrödinger equation described by hyperspherical coordinates, Int. J. Quan. Chem. 45 (1993) 385.

       

       J. Raynal and J. Revai, Transformation coefficients in the hyperspherical approach to the three-body problem, Nuovo Cimento, 68 (1970) 612-622.

       

       J. Ackermann, Global and local properties of the S states of the dtμ molecular ion: A finite-element study, Phys. Rev. A 57 (1998) 4201-4203.

       

       C. D. Lin, Classification and supermultiplet structure of doubly excited states, Phys. Rev. A. 29 (1984) 1019-1033.

       

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