Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics

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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 ψ{\psi} the quantity AA is equal to λ{\lambda} if ψ{\psi} is an eigenfunction of the operator O^(A){\hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{\hat{O}((A-\lambda)2)\psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkin’s mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){\hat{O}((A-\lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{\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 λ{\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  ±iD^{\pm i \sqrt{\langle \hat{D} \rangle}}.

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1. Introduction The Kuryshkin-Wodkiewicz quantum mechanics [1] is an example of a dissipative quantum system. The quantum part of the measuring device is the ‘environment of an open quantum system’. In the process of quantum measurement, an open quantum system interacts with its ‘environment’. We study the result of this interaction [2]-[12]. Therefore, wave vectors must have a finite lifetime, inversely proportional to the imaginary part of eigenvalues. In this version of quantum mechanics, the von Neumann quantization rule was abandoned and observable quantities are assigned to pseudo-differential operators, not necessarily self-adjoint. Therefore, the appearance of the imaginary part of the eigenvalues is not surprising. However, our studies of hydrogen-like atoms have shown that the operator corresponding to the Hamiltonian is essentially self-adjoint, so its discrete spectrum turned out to be real [13], [14]. This is quite surprising, since the von Neumann rule can be derived from general considerations, if we assume that the relation between the quantities

About the authors

Alexander V. Zorin

Peoples’ Friendship University of Russia (RUDN University)

ORCID iD: 0000-0002-5721-4558

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research

ORCID iD: 0000-0001-6541-6603

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research

Author for correspondence.
ORCID iD: 0000-0002-1856-4643

Doctor of Physical and Mathematical Sciences, Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation


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