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Доклады Академии наук

Randomizes hamiltonian mechanics

Yu. N. Orlov, Vsevolod Zhanovich Sakbaev, Oleg Georgievich Smolyanov

Published Jun 28, 2019• 1 citation
Publisher

Abstract

Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.

DOI
10.31857/s0869-56524866653-658

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Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.
Abstract Hamiltonian mechanics determined by time-dependent random Hamiltonian functions is called randomized Hamiltonian mechanics. The corresponding Hamiltonian systems are called random. Feynman formulas for random Hamiltonian systems are obtained. It is shown that these formulas describe solutions of a Hamiltonian equation whose Hamiltonian is the mean value of the random Hamiltonian function. Analogous results are obtained for random quantum systems (which are known to be infinite-dimensional random Hamiltonian systems). The random quantum Hamiltonians can be used to describe the so-called open quantum systems (which are part of some lager quantum systems).
Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton's equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.