Principles of the first and second kind of balance in a varying-parameters method for eigenvalue problems in quantum mechanics

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Abstract
In this paper a method for eigenvalue problems in quantum mechanics is developed. Principles of the first and second kinds of balance in a varying-parameters method take advantage of the standard perturbation theory and the standard variational principle. According to this method, we not only obtain the best approach for obtaining eigenvalues and wave functions, but we also determine structures of a quantum system. The extended Hamiltonian in a varying-parameters method is different from a standard Hamiltonian. The parameters are inserted into a Hamiltonian by adding and subtracting appropriate terms which contain the essential parameters. Thus it becomes possible to study the inner structure of a quantum system by applying principles of balance. In order to interpret the physical meanings of balance parameters, several examples are described. Applications are also made to the helium atom, heliumlike ions, and the lithium atom. We theoretically predict energies and structure parameters and obtain good agreement with experimental data, especially for high-orbit electrons (clearly this is a character of perturbation theory). It is quite interesting that the theoretical predictions of energy levels of parahelium in the S state are lower than the energy levels of orthohelium and that theoretical predictions of singlet (triplet) states are close to the experimental data of triplet (singlet) states. It would seem that the experimental data of triplet and singlet are reversed. Is that possible?