On a Method of Calculating Quantum Distribution Functions

A. V. Zorin, N. P. Tretyakov

Abstract In the Maple computer algebra system, an algorithm is implemented for symbolic and numerical computations for finding the transition probabilities for hydrogen-like atoms in quantum mechanics with a nonnegative quantum distribution function (QDF). Quantum mechanics with a nonnegative QDF is equivalent to the standard theory of quantum measurements. However, the presence in it of a probabilistic quantum theory in the phase space gives additional possibilities for calculating and interpreting the results of quantum measurements. The methods of computer algebra seem to be necessary for the relevant calculations. The calculation of the matrix elements of operators is necessary for determining the energy levels, oscillator strengths, and radiation transition parameters for atoms and ions with an open shell. Transition probabilities are calculated and compared with experimental data. They are calculated using the Galerkin method with the Sturm functions of the hydrogen atom as coordinate functions. The verification of the model showed good agreement between the calculated and experimentally measured transition probabilities.

Nils Dalarsson, Mariana Dalarsson, Leonardo Golubović

Joachim J. Włodarz, Janusz Nowakowski

Abstract The quantum (ħ =1) and semiclassical (ℏ → 0) Wigner distribution functions were calculated for some anharmonic oscillators using a direct phase‐space variational method.

Eugene Paul Wigner

А. В. Зорин, А. В. Зорин

The method of calculating probabilities of radiation transitions in hydrogen-like atoms in quantum mechanics with nonnegative distribution function is proposed. Galerkin method using Sturmian functions of the hydrogen atom as basis functions allows to reduce the computations to algebraic operations with matrix elements calculated analitically.