Abstract
We have constructed the three-body permutation symmetric O(6) hyperspherical harmonics which can be used to solve the non-relativistic three-body Schrödinger equation in three spatial dimensions. We label the states with eigenvalues of the \(U(1) \otimes SO(3)_{rot} \subset U(3) \subset O(6)\) chain of algebras and we present the corresponding \(K \le 4\) harmonics. Concrete transformation properties of the harmonics are discussed in some detail.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The mixed symmetry representation of the \(S_3\) permutation group being two-dimensional, there are two different state vectors (hyperspherical harmonics) in each mixed permutation symmetry multiplet, usually denoted by \(M_{\rho }\) and \(M_{\lambda }\).
References
T.H. Gronwall, Phys. Rev. 51, 655 (1937).
J.H. Bartlett, Phys. Rev. 51, 661 (1937).
L.M. Delves, Nucl. Phys. 9, 391 (1958); ibid. 20, 275 (1960).
F.T. Smith, J. Chem. Phys. 31, 1352 (1959); F.T. Smith, Phys. Rev. 120, 1058 (1960); F.T. Smith, J. Math. Phys. 3, 735 (1962); R.C. Whitten and F.T. Smith, J. Math. Phys. 9, 1103 (1968).
Yu.A. Simonov, Sov. J. Nucl. Phys. 3, 461 (1966) [Yad. Fiz. 3, 630 (1966)].
T. Iwai, J. Math. Phys. 28, 964, 1315 (1987).
I. Salom and V. Dmitrašinović, “Permutation-symmetric three-particle hyper-spherical harmonics based on the \(S_3 \otimes SO(3)_{rot} \subset U(1) \otimes SO(3)_{rot} \subset U(3) \subset O(6)\) chain of algebras”, in preparation (2015).
N. Barnea and V.B. Mandelzweig, Phys. Rev. A 41, 5209 (1990).
V.A. Nikonov and J. Nyiri, International Journal of Modern Physics A, Vol. 29, No. 20, 1430039 (2014).
V. Dmitrašinović, T. Sato and M. Šuvakov , Phys. Rev. D 80, 054501 (2009).
Acknowledgements
This work was financed by the Serbian Ministry of Science and Technological Development under grant numbers OI 171031, OI 171037 and III 41011.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Salom, I., Dmitrašinović, V. (2016). Permutation-Symmetric Three-Body O(6) Hyperspherical Harmonics in Three Spatial Dimensions. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. LT 2015. Springer Proceedings in Mathematics & Statistics, vol 191. Springer, Singapore. https://doi.org/10.1007/978-981-10-2636-2_31
Download citation
DOI: https://doi.org/10.1007/978-981-10-2636-2_31
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-2635-5
Online ISBN: 978-981-10-2636-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)