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 Physics , 2000, DOI: 10.1143/PTP.104.575 Abstract: We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shing\=u. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.
 Physics , 1995, DOI: 10.1103/PhysRevA.53.3767 Abstract: In the Heisenberg picture, the generalized invariant and exact quantum motions are found for a time-dependent forced harmonic oscillator. We find the eigenstate and the coherent state of the invariant and show that the dispersions of these quantum states do not depend on the external force. Our formalism is applied to several interesting cases.
 Physics , 1998, DOI: 10.1088/0305-4470/31/39/002 Abstract: 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 states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained.
 I. O. Vakarchuk Physics , 2005, DOI: 10.1088/0305-4470/38/34/010 Abstract: The Dirac equation has been studied in which the Dirac matrices $\hat{\boldmath$\alpha$}, \hat\beta$ have space factors, respectively $f$ and $f_1$, dependent on the particle's space coordinates. The $f$ function deforms Heisenberg algebra for the coordinates and momenta operators, the function $f_1$ being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schr\"odinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomnb field with a linear dependence of the $f$ function on the distance $r$ to the force centre and the inverse dependence on $r$ for the $f_1$ function has been found.
 Sang Pyo Kim Physics , 2003, Abstract: We investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant operators, known as the Ermakov-Lewis invariants, in terms of a complex classical solution, from which the evolution operator is derived, and obtain the Heisenberg position and momentum operators. Physical quantities such as correlation functions are calculated using both the invariant operators and Heisenberg operators.
 Physics , 2006, DOI: 10.1063/1.2222273 Abstract: We describe a method for calculating dynamical spin-spin correlation functions in the finite isotropic and anisotropic antiferromagnetic Heisenberg models. Our method is able to produce results with high accuracy over the full parameter space.
 Physics , 2010, DOI: 10.1103/PhysRevD.82.125031 Abstract: One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron Zitterbewegung: it introduces intraband frequency components into the motion, influences all the frequencies and makes the motion stationary (not decaying in time) in case of the 2+1 Dirac equation. Finally, simulations of the 2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.
 Physics , 2005, Abstract: The quasi-Heisenberg picture of minisuperspace model is considered. The The quasi-Heisenberg picture of minisuperspace model is considered. The suggested scheme consists in quantizing of the equation of motion and interprets all observables including the Universe scale factor as the time-dependent (quasi-Heisenbeg) operators acting in the space of solutions of the Wheeler--DeWitt equation. The Klein-Gordon normalization of the wave function and corresponding to it quantization rules for the equation of motion allow a time-evolution of the mean values of operators even under constraint H=0 on the physical states of Universe. Besides, the constraint H=0 appears as the relation connecting initial values of the quasi-Heisenbeg operators at $t=0$. A stage of the inflation is considered numerically in the framework of the Wigner--Weyl phase-space formalism. For an inflationary model of the chaotic inflation'' type it is found that a dispersion of the Universe scale factor grows during inflation, and thus, does not vanish at the inflation end. It was found also, that the by hand'' introduced dependence of the cosmological constant from the scale factor in the model with a massless scalar field leads to the decrease of dispersion of the Universe scale factor. The measurement and interpretation problems arising in the framework of our approach are considered, as well.
 Physics , 2008, Abstract: The unmodified Heisenberg-Pauli canonical formalism of quantum field theory applied to a self-interacting scalar boson field is shown to make sense mathematically in a framework of generalized functions adapted to nonlinear operations. The free-field operators, their commutation relations, and the free-field Hamiltonian operator are calculated by a straightforward transcription of the usual formalism expressed in configuration space. This leads to the usual results, which are essentially independent of the regularization, with the exception of the zero-point energy which may be set to zero if a particular regularization is chosen. The calculations for the self-interacting field are more difficult, especially because of the well-known problems due to the unboundness of the operators and their time-dependent domains. Nevertheless, a proper methodology is developed and a differentiation on time-dependent domains is defined. The Heisenberg equations and the interacting-field equation are shown to be mathematically meaningful as operator-valued nonlinear generalized functions, which therefore provide an alternative to the usual Bogoliubov-Wightman interpretation of quantized fields as operator-valued distributions. The equation for the time-evolution operator is proved using two different methods, but no attempt is made to calculate the scattering operator, and the applications to perturbation theory are left to a subsequent report.
 Physics , 2007, Abstract: Time dependent entropy of constant force motion is investigated. Their joint entropy so called Leipnik's entropy is obtained. The main purpose of this work is to calculate Leipnik's entropy by using time dependent wave function which is obtained by the Feynman path integral method. It is found that, in this case, the Leipnik's entropy increase with time and this result has same behavior free particle case.
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