Hydrogen-like atom with nonnegative quantum distribution function

Abstract

Among numerous approaches to probabilistic interpretation of conventional quantum mechanics (CQM), the closest to N. Bohr’s idea of the correspondence principle is the Blokhintzev-Terletsky approach of the quantum distribution function (QDF) on the coordinate-momentum (q, p) phase space. The detailed investigation of this approach has led to the correspondence rule of V.V. Kuryshkin parametrically dependent on a set of auxiliary functions. According to investigations of numerous authors, the existence and the explicit form of QDF depends on the correspondence rule between classical functions A(q, p) and quantum operator A. At the same time, the QDF corresponding to all known quantization rules turns out to be alternating in sign or overly complex valued. Finally nonexistence of nonnegative QDF in CQM was proved. On the other hand, from this follows the possibility to construct quantum mechanics where a nonnegative QDF exists. We consider a certain set of auxiliary functions to construct explicit expressions for operators O(H) for the hydrogen atom. Naturally, these operators differ from the related operator Ĥ in CQM, so that spherical coordinates are no longer separable for a hydrogen-like atom in quantum mechanics with nonnegative QDF.