On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.     

On the exact discretization of the classical harmonic oscillator equation

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.     

A map between the coulomb and the harmonic oscillator systems in a higher-dimensional space

We present an extended transformation method for mapping between higher-dimensional spaces. We report solutions of the Schrödinger equation for the Green’s function in the form of bound states and scattering states. We discuss the normalizability of the bound-state solution of a generated exactly solvable potential.     

On Gibbs quantum and classical particle systems with three-body forces

For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 976–996, July, 2006.     

Equation of motion for a radiating charged particle in classical electrodynamics

We propose classical equations of motion for charged particles in an electromagnetic field. These are general formulas for the particle acceleration that take the radiation-induced deceleration into account and contain no second derivatives of the particle velocity. In several particular cases considered, the new equations yield results coinciding with those known in the literature and experimentally verified. We show that in the range of ultrahigh energies, classical electrodynamics does not lead to inherent inconsistencies and in principle allows particle motion with energies exceeding the Pomeranchuk limit.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 112–130, April, 2005.     

Quantum and classical fidelity for singular perturbations of the inverted and harmonic oscillator

Let us consider the quantum/versus classical dynamics for Hamiltonians of the form

(0.1)     

Quantum hydrodynamics of particle systems with coulomb interaction and quantum bohm potential

The quantum hydrodynamics of N interacting particles with Coulomb interaction in an external electromagnetic field can be described by the field equations for the microscopic dynamics in the physical space. Macroscopic hydrodynamic equations are obtained by local averaging. Quantum corrections to the hydrodynamic equations are due to the multiparticle quantum Bohm potential. Specific properties of Fermi- and Bose-system hydrodynamics are investigated. The Cauchy-type integral for the quantum system and the corresponding one-particle Schrödinger equation are found under the standard classical hydrodynamic assumptions.     

On the mean-square stability for a harmonic oscillator with random parameter

Sufficient mean-square stability conditions of a harmonic oscillator whose random parameter is an Ornstein-Uhlenbeck process are obtained.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1276–1278, September, 1992.     

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.     

On the hidden parameter in quantum and classical physics

Mathematical Notes -     

On simulating the quantum and classical branching programs

The complexity classes defined on the basis of branching programs are considered. Some basic relations are established between the complexity classes defined by the probabilistic and quantum branching programs (measure-once, as well as measure-many), computing with bounded or unbounded error. To prove these relations, we developed a method of “linear simulation” of a quantum branching program and a method of “quantum simulation” of a probabilistic branching program.     

On the Weyl symbol of the resolvent of the harmonic oscillator

We compute the Weyl symbol of the resolvent of the harmonic oscillator and study its properties.     

Stability and bifurcation in a harmonic oscillator with delays

We consider a harmonic oscillator with delays. Linear stability is investigated by analyzing the associated characteristic transcendental equation. The bifurcation analysis of the equation shows that Hopf bifurcation can occur as the delay τ (taken as a parameter) crosses some critical values. The direction and stability of the Hopf bifurcation are considered by using the normal form theory due to Faria and Magalhães. An example is given to explain the results. Numerical simulations support our results.     

Quantum trajectories of an oscillator interacting with an electromagnetic field, a classical force, and a heat bath

We consider a solvable problem describing the dynamics of a quantum oscillator interacting with an electromagnetic field, a classical force, and a heat bath. We propose a general method for solving Markovian master equations, the method of quantum trajectories. We construct the stochastic evolution operator involving the stochastic analogue of the Baker-Hausdorff formula and calculate the system density matrix for an arbitrary initial state. As a physical application, we evaluate the influence of the environment at a finite temperature on the accuracy of measuring a weak classical force by the interference method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 444–459, March, 2009.