Abstract
We present symbolic algorithms realized in REDUCE 3.6 for evaluation of eigenvalues and eigenfunctions of the 3-D and 2-D hydrogen atoms in weak uniform electric fields. Algebraic perturbation theory schemes are built up using the irreducible representations of the dynamical symmetry algebras so(4,2) and so(3,2), which are connected by the tilting transformations with ‘wave functions of the 3-D and 2-D hydrogen atoms. Such a construction is based on a representation of the unperturbed Hamiltonian and polynomial perturbation operator via generators of the algebra. It was done without an assumption on the separation of independent variables of the perturbation operator and without using fractional powers of the parabolic quantum numbers in recurrence relations determining the effects of generators of the algebra on the corresponding basis. The efficiency of the proposed schemes and algorithms is demonstrated by calculations of coefficients of the Stark effect perturbations series for the hydrogen atoms with arbitrary parabolic quantum numbers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Malkin, LA., Man’ko, V.I.: Dynamical symmetry and coherent satates of quantum systems, Nauka, Moscow, 1979 (in Russian).
Adams, B.G., Cizek, J., and Paldus, J.: Lie algebraic methods and their applications to simple quantum systems. In: Advances in Quantum Chemistry, Per-Olov Lowdin (Ed.), Academic Press, New York, 18 (1988), pp. 1–85.
Cisneros, A., and McIntosh, H.V.: Symmetry or Two-Dimension Hydrogen Atom. J. Math. Phys. 10 (1969) 277–286.
Engerfield, M.J.: Group Theory and the Coulomb Problem, Monash university, Victoria (1972).
Adams, B.G.: Unified treatment of high-order perturbation theory for Stark effect in a two- and three-dimensional hydrogen atom. Phys. Rev. A. 46 (1992) 4060–4064.
Kadomtsev, M.B. and Vinitsky, S.I.: Perturbation theory within the 0(4,2) group for hydrogen atom in the field of distant charge. J. Phys. A: Math. Gen. 18 (1985) L689–L695.
Silverstone, H.J.: Perturbation theory of the Stark effect in hydrogen to arbitrarily high order. Phys. Rev. A 18 (1978) 1853–1864.
Silverstone, H.J., Moats, R.K.: Practical recursive solution of degenerate RayleighShroedinger perturbation theory and application to high-order calculation of the Zeeman-effect in hydrogen. Phys. Rev. A 23 (1981) 1645–1654.
Abrashkevich, A.G., Puzynin, LV., Vinitsky, S.I.: ASYMPT: a program for calculating asymptotics of hyperspherical potential curves and adiabatic potentials. Computer Physics Communications 125 (2000) 259–281.
Courant, R. and Hilbert, D.: Methods of Mathematical Physics, Interscience publishers, New-York, London, 1953.
Lebedev, N.N.: Special Functions and Their Application, GITTL, Moscow, 1953 (in Russian).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gusev, A., Samoilov, V., Rostovtsev, V., Vinitsky, S. (2000). Symbolic Algorithms of Algebraic Perturbation Theory for a Hydrogen Atom: the Stark Effect. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57201-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-57201-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41040-9
Online ISBN: 978-3-642-57201-2
eBook Packages: Springer Book Archive