Symbolic Algorithms of Algebraic Perturbation Theory for a Hydrogen Atom: the Stark Effect

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.