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We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus…
Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be…
There are discussed the exact solution of the time--dependent Schr\"{o}dinger equation for a damped quantum oscillator subject to a periodical frequency delta--kicks describing squeezed states which are expressed in terms of Chebyshev…
A modified quantum kinetic equation which takes account of the noninertial features of rotating frame is proposed. The vector and axial-vector field components of the Wigner function for chiral fluids are worked out in a semiclassical…
For $N$-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number…
A Wigner-Klein-Kramers equation is proposed, which merges relativistic, quantum and thermo dynamics. The relativistic effect on quantum Brownian motion is studied via the Breit-Fermi Hamiltonian applied into a dissipative Madelung…
We review some aspects of the quantization of the damped harmonic oscillator. We derive the exact action for a damped mechanical system in the frame of the path integral formulation of the quantum Brownian motion problem developed by…
By using the Lewis-Riesenfeld theory and the invariant-related unitary transformation formulation, the exact solutions of the {\it time-dependent} Schr\"{o}dinger equations which govern the various Lie-algebraic quantum systems in atomic…
We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard…
We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the…
Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum…
Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrodinger equations of Lie type and we show how these methods explain…
In this paper, the principles of the general relativity are used to formulate quantum wave equations for spin-0 and spin-1/2 particles. More specifically, the equations are worked in a Schwarzschild-like metric. As a test, the hydrogen atom…
A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to…
It is shown that the well-known relativistic correction of quantum Hamiltonian that is present in textbooks appears after quantization of oversimplified relativistic kinetic energy decomposition. Using the proper expression one obtains the…
In this contribution a path integral approach for the quantum motion on three-dimensional spaces according to Koenigs, for short``Koenigs-Spaces'', is discussed. Their construction is simple: One takes a Hamiltonian from three-dimensional…
We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particles systems, we derive the analogous results for the…
The dissipative quantum electromagnetics is introduced in a comprehensive manner as a field-matter-bath coupling problem. First, the matter is described by a cluster of Lorentz oscillators. Then the Maxwellian free field is coupled to the…
We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system…
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the…