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Symmetrization of Hyperspherical Basis for Few-Body Systems

Exploring symmetrization in hyperspherical basis for 4, 5, and 6-body systems using Parentage Scheme of Symmetrization. Deriving transformation coefficients and Young operators for identical particle systems.

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Symmetrization of Hyperspherical Basis for Few-Body Systems

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  1. 0 Constructing Wave Functions for Few-Body Systems in a Hyperspherical Basis Using Parentage Scheme of Symmetrization Lia Leon Margolin, Ph.D. Associate Professor of Mathematics Marymount Manhattan Collage, New York. NY

  2. Problem statement • Investigating few-body systems with identical particles in a hyperspherical basis yields the problem of obtaining symmetrized hyperspherical functions from functions with arbitrary quantum numbers. • This article solves the problem of hyperspherical basis symmetrization for four-,five- ,and six- body systems using Parentage Scheme of Symmetrization. JibutiR.I., Krupennikova N.B., Chachanidze-Margolin L.L. Few Body Systems, 4, 151, (1988) Margolin L.L. Journal of Physics, 343, 1 (2012) Margolin L.L. EPJ Web of Conferences 66, 09013 (2014)

  3. Presentation Outline • Hyperspherical Functions (HF) for the systems with different particles • Different Configurations for Four-Five and Six particle systems • Transformation Coefficients for N=3,4,5,6 Body Systems • Transformations of N-Particle Systems with N-1 Identical Particles when N=4,5,6 • Parentage Scheme of Symmetrization for N=4,5,and 6 Body Systems • Constructing Young operators acting on N=4,5,and 6 Body HF symmetrized with respect to N-1 particles • Finding Parentage coefficients for N=3,4,5,and 6 Body Systems • Constructing Fully symmetrized HF with specific quantum numbers for N=3,4,5,and 6 Body Systems • Conclusion

  4. Abstract • Hyperspherical basis symmetrization for four-,five- and six- body systems using Parentage Scheme of Symmetrization. • Parentage coefficients corresponding to the [4], [31], [22], [211], representations of S_4 groups, [5], [311], [221], [2111], [11111] representations of S_5 groups, and [42] and [51] representations of S_6 groups are obtained, • Young operators, acting on N = 4,5,6 body hyperspherical functions symmetrized with respect to (N-1) particles, are derived. The connection between the transformation coefficients for the identical particle systems and the parentage coefficients is demonstrated and the corresponding formulas are introduced.

  5. Four Particle System Configurations 0 • (3+1) Configuration (2+2) Configuration Jibuti R.I., Krupennikova N.B., Chachanidze-Margolin L.L. Few Body Systems, 4, 151, (1988) Margolin L.L. Journal of Physics, 343, 1 (2012)

  6. 0 (3+1) Configuration for

  7. Transformation from Jacobi to Four Body Hyperspherical Coordinates

  8. Four Particle HF in Nine Dimensional Space of Jacobi Vectors

  9. Four Body HF with L=l12 +l3

  10. Transformations of four-body hyperspherical functions

  11. Importance of Recurrence Method • The transformations of Hyperspherical functions become sufficiently complex when number of particles in the system increases • For four and more particles kinematic rotations (KR) include both particle permutations and transitions from one configuration to another • Finding (KR) coefficients for four particle systems using general formula is extremely difficult and is practically impossible for the systems with five and more particles • Recurrent method allows to obtain KR coefficients for the systems with any number of particles

  12. Five Particle System Configurations 0 (4+1) Configuration (2+2+1) Configuration (3+2) Configuration Krupennikova N.B., Chachanidze-Margolin L.L. Proc. X –Europ. Conference on Few-Body Physics (1990) Margolin, L.L. EPJ Web of Conferences 66, 09013 (2014)

  13. Five Body Hyperspherical Basis

  14. Five Body Kinematic Rotations

  15. (5+1) Configuration for Six Body Systems

  16. Six Body Hyperspherical Functions w

  17. Four body Hyperspherical Functions symmetrized with respect to three identical particles Margolin L.L. Journal of Physics, 343, 1 (2012)

  18. Transformation matrix for four particle systems with three identical particles

  19. Transformations of Four Body HF with three identical particles 0 Margolin L.L. Journal of Physics, 343, 1 (2012)

  20. Transformation Coefficients of four-body HF with three identical particles Margolin, L.L. EPJ Web of Conferences 66, 09013 (2014

  21. Parentage scheme of Symmetrization . N=3 we have [3],[21] and [111] N=4 we have [4],[22],[31],[211] N=5 we have [5],[41],[221],[311],[2111],[11111] N=6 we have [6],[51],[42],[411],[2211],[3111],[111111]

  22. Young Operators of Group S3

  23. Young Operators of Group S4

  24. Young Operators of Group S5

  25. Young Operators of Group S5

  26. Young Operators of Group S5 (cont.)

  27. Young Operators of Group S5 (cont.)

  28. Young Operators of Group S5 (cont.)

  29. Young Operators Acting on Six body Functions . . . . .

  30. : Young Operators of Six Body Systems . . . . . . . . . .

  31. Parentage Scheme of Symmentrization

  32. Parentage Coefficients for Three Body Systems

  33. Parentage Coefficients for Four Body Systems

  34. Parentage Coefficients for Five Body Systems

  35. Parentage Coefficients for six body HF where coefficients

  36. Constructing Four Body Symmetrized Hyperspherical Basis

  37. Four Body Symmetrization Coefficients

  38. Symmetrized three and four Body HF N=3: N=4:

  39. Symmetrized Five Body HF

  40. Symmetrized Six Body HF

  41. Conclusions • Recurrence method allows to find unitary transformation coefficients for the N particles systems of various types. • According to the PSS, N-body hyperspherical functions corresponding to the representation of the N-particle permutation group Sn can be obtained by finding parentage coefficients and constructing linear combinations of the N-particle functions corresponding to the irreducible representations of N-1 particle permutation group Sn-1. • According to the PSS we need to construct N-body functions symmetrized with respect to N-1 particles first and then calculate parentage coefficients acting on these functions.

  42. Talks and Publications L Margolin EPJ Web of Conferences 66, 09013 (2014) L.L. Margolin, Sh. Tsiklauri, “Kinematic rotations of N-particle hyperspherical basis,” August 2006, 8pp, e-Print Archive:nucl-th/0608001 ia Leon Margolin 2012 J. Phys.: Conf. Ser.343 01207 Chachanidze-Margolin L.L, “Development of the Hyperspherical Function Method in Impulse Representation applicable for the Microscopic Investigations of Various Nuclear and Hypernuclear Systems”, Proc. Of the VI-th Joint International AMS-SMM Conference, ”, Houston, TX, May 13-15, 2004 Chachanidze-Margolin L.L., “The Kinematical Rotations of N-particle Hyperspherical basis. Construction of Symmetrized Basic Hyperspherical Functions”, 2003 Spring Eastern Sectional Meeting of American Mathematical Society (AMS), New York, New York, April 12-14

  43. 0 Kinematic Rotations of N-particle Hyperspherical Basis at K_{N}=2, L_{N}=0. (1)

  44. Three and Four Body Systems

  45. Orthonormal Hyperspherical basis

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