Narrow Results

We obtain the Schrödinger wave functions of a generalized pendulum under time-dependent gravitation by making use of the Lewis and Riesenfeld invariant method. As an example, we consider a generalized pendulum with constant gravitation and exponentially increasing mass. We also present a canonical approach to the generalized time-dependent pendulum.

In this work, we analyze the quantum dynamics of a generalized pendulum with a time-varying mass increasing exponentially and constant gravitation. By using Lewis–Riesenfeld invariant approach and Fock states, we solve the time-dependent Schrödinger equation for this system and write its solutions in terms of solutions of the Milne–Pinney equation. We also construct coherent states for the quantized pendulum and use both Fock and coherent states to investigate some important physical proprieties of the quantized pendulum such as eigenvalues of the angular displacement and momentum, their quantum variances as well as the respective uncertainty principle. Finally, we derive the geometric, dynamical and Berry phases for the time-dependent generalized pendulum.

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with ${𝒫}{𝒯}$ symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having ${𝒫}{𝒯}$ symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schrödinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted harmonic oscillator. As a special case, we consider a generalized inverted oscillator with constant frequency and exponentially increasing mass.

In this paper, we study the generalized harmonic oscillator with arbitrary time-dependent mass and frequency subjected to a linear velocity-dependent frictional force from classical and quantum points of view. We obtain the solution of the classical equation of motion of this system for some particular cases and derive an equation of motion that describes three different systems. Furthermore, with the help of the quantum invariant method and using quadratic invariants we solve analytically and exactly the time-dependent Schrödinger equation for this system. Afterwards, we construct coherent states for the quantized system and employ them to investigate some of the system's quantum properties such as quantum fluctuations of the coordinate and the momentum as well as the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary system. Finally, we evaluate the dynamical and Berry phases for three special cases and surprisingly find identical expressions for the dynamical phase and the same formulae for the Berry's phase.

By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.

One of the most widely used methods for preprocessing binary images is thinning. The popularity of this method rests on the fact that considerable data reduction is achieved while retaining “essential” properties of the original image. Moreover, topological features, which cannot be verified by a genuinely parallel method (by Minsky and Papert²²) are more easily treated in thinned images. For these reasons, many articles on this topic were published in the literature. Most of them are concerned with modifications of existing methods in order to yield “nicer” results out of the thinning process. Also, many results of numerical experiments are available in different publications.

The aim of this paper is to show that the quite natural requirement of invariance of the results obtained by thinning leads nearly automatically to a method proposed by the authors in different publications. Moreover, this method is genuinely parallel and well-defined so that it is possible to investigate it theoretically.

The practical feasibility of the method is also discussed.

One of the most widely used methods for preprocessing binary images is thinning. The popularity of this method rests on the fact that considerable data reduction is achieved while retaining “essential” properties of the original image. Moreover, topological features, which cannot be verified by a genuinely parallel method (by Minsky and Papert²²) are more easily treated in thinned images. For these reasons, many articles on this topic were published in the literature. Most of them are concerned with modifications of existing methods in order to yield “nicer” results out of the thinning process. Also, many results of numerical experiments are available in different publications.

The aim of this paper is to show that the quite natural requirement of invariance of the results obtained by thinning leads nearly automatically to a method proposed by the authors in different publications. Moreover, this method is genuinely parallel and well-defined so that it is possible to investigate it theoretically.

The practical feasibility of the method is also discussed.