Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics
| Autor: | Alexander V. Zorin, Mikhail D. Malykh, Leonid A. Sevastianov |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Discrete and Continuous Models and Applied Computational Science, Vol 30, Iss 2, Pp 139-148 (2022) |
| Druh dokumentu: | article |
| ISSN: | 2658-4670 2658-7149 |
| DOI: | 10.22363/2658-4670-2022-30-2-139-148 |
| Popis: | One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){\hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state {\psi} the quantity AA is equal to {\lambda} if {\psi} is an eigenfunction of the operator O^(A){\hat{O}(A)}. In this case, the variance O^((A-)2){\hat{O}((A-\lambda)2)\psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkins mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-)2){\hat{O}((A-\lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)+2E^{\hat{O}(A2)-2\hat{O}(A)\lambda + \lambda 2 \hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{\hat{D}(A) = \hat{O}(A2) - \hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue {\lambda} of the operator O^(A){\hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by i⟨D^⟩{\pm i \sqrt{\langle \hat{D} \rangle}}. |
| Databáze: | Directory of Open Access Journals |
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