Abstract
A new method for calculation of non-relativistic energy spectrum of Coulomb three-body systems with two identical particles has been developed. The novelty of the method is the introduction of an expansion of the wave function on harmonic oscillator (HO) functions with different sizes in the Jacobi coordinates instead of only one unique size parameter in the traditional approach. The method presented obeys the principles of antisymmetry and translational invariance. The theoretical formulation has been illustrated by evaluation of ground state energies of a number of Coulomb three-body systems with two identical particles for zero HO excitation energy. The analytical solution of this problem in case of only one size parameter has been derived. The obtained results show significant advantage of the base with different sizes over the traditional approach for investigation of the bound state problem of quantum systems.
Highlights
The theoretical description of quantum systems should obey the principles of antisymmetry and translational invariance
The proposed formalism consistently outlines the principles of antisymmetrization and translational invariance
Compared with the traditional approach, the method is based on an expansion of the wave function on harmonic oscillator (HO) functions with different sizes in the Jacobi coordinates
Summary
The theoretical description of quantum systems should obey the principles of antisymmetry and translational invariance. It should be pointed that there exist a wide variety of the methods that start with antisymmetrization in single particle variables and only project out the excited states of the centrum of mass motion, ensuring the translational invariance of the calculations (see, e.g., [12] and references therein) This approach may cope with enormous technical problems since the projection procedure should be performed in very large single particle bases even for systems with small number of particles. Three-body Coulomb systems with two identical particles include the two electron atoms/ions, a variety of diatomic molecular ions, e.g. hydrogen and its isotops, as well as the exotic mesonic systems or positronium ion It should be noted, that there exist a lot of different highly accurate approximations of wave functions and energy levels for three-body Coulomb systems states that are constructed by exponential expansion based variational approach (see, e.g., [13] and references therein).
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