Abstract
The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. The program is written up in the language MAPLE. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch-Gordan coefficients, etc. is proposed. Operators of observables are calculated on the basis of the QDF quantization rule. According to the Ritz method, eigenvalues of Ritz matrices represent spectral values of the quantity under investigation, i.e. energy. As an example, energy levels of hydrogen-like atoms are calculated and compared with experimental data retrieved from the NIST Atomic Spectra Database Levels Data. It turns out that this theory seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics.
Highlights
In [1] several computational techniques for investigation of characteristics of atomic structures were proposed
In the case of n = 1, l = 0, sturmian and usual Coulomb wave functions coincide exactly, but as n grows, the amplitude and number of harmonics of sturmian functions differ from the Coulomb ones
The main goal of the program is a numerical calculation of the spectrum of hydrogen and some alkali metals, using the Ritz method with 91 × 91 matrices. This corresponds to the use of the basis of functions with the values of the principal quantum number n = 1, 2, 3, 4, 5, 6
Summary
In [1] several computational techniques for investigation of characteristics of atomic structures were proposed. Pp. 343–356 consists in improving Hartree–Fock-like wavefunctions by introducing configuration interaction (CI) while using the usual dipole operator for the oscillator strength This approach was used by Burke et al [6] for Be, Zare [7] and Weis [8] for Mg, and Friedrich and Trefftz [9] for Ca and Ba. An alternative approach was adopted in [10], which uses an effective operator in addition to the common dipole operator and in so doing keeps a rather simple structure of corresponding wave functions. The weak point in this theory lays in the fact that until the present time no truly extensive investigations of properties of real quantum systems have been conducted This is due to the much more complex structure of the QDF theory as compared with traditional quantum mechanics. This article sums up the many years of work on the creation of a package of programs and poses challenges for future research
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