A class of exactly solvable Schrödinger equation with moving boundary condition

Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary condition.     

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.     

Tunneling in a “breathing” double well: Adiabatic and antiadiabatic limits and tunneling suppression

Tunneling in a piecewise harmonic potential coupled to a harmonic oscillator is considered by means of the path integral technique. The reduced propagator for the tunneling particle is calculated explicitly and the tunneling splitting is found in semiclassical approximation. The result holds for arbitrary values of the parameters of the system. From this the adiabatic and antiadiabatic approximations are obtained as particular cases and compared with the results obtained differently. The limit of a strong interaction is also considered. It is found that for strong interaction or equivalently for the harmonic frequency tending to zero the preexponential factor in the tunneling splitting tends to zero which results in a suppression of tunneling. Implications of this result for tunneling in a more general potential are discussed.     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Infinite quantum well: A coherent state approach

A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.     

New approach for solving the Wigner function from the Husimi function in the two-mode entangled state representation

We introduce a new method to calculate the Wigner function when its corresponding Husimi function is given. A new formula is derived for calculating conveniently the Wigner function in two-mode entangled state representation. As application, we derive Wigner functions of some quantum states, such as two-mode entangled state, the electron's two-mode squeezed canonical coherent state, and the electron's coordinate eigenstate.     

Quantum-classical correspondence of the relativistic equations

According to the Heisenberg correspondence principle, in the classical limit, quantum matrix element of a Hermitian operator reduces to the coefficient of the Fourier expansion of the corresponding classical quantity. In this article, such a quantum-classical connection is generalized to the relativistic regime. For the relativistic free particle or the charged particle moving in a constant magnetic field, it is shown that matrix elements of quantum operators go to quantities in Einstein’s special relativity in the classical limit. Especially, matrix element of the standard velocity operator in the Dirac theory reduces to the classical velocity. Meanwhile, it is shown that the classical limit of quantum expectation value is the time average of the classical variable.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Formal solutions of stargenvalue equations

The formal solution of a general stargenvalue equation is presented, its properties studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. Our approach deals with discrete and continuous spectra in a unified fashion and includes a systematic treatment of nondiagonal stargenfunctions. The formalism is used to obtain a complete formal solution of Wigner quantum mechanics in the Heisenberg picture and to write a general formula for the stargenfunctions of Hamiltonians quadratic in the phase space variables in arbitrary dimension. A variety of systems is then used to illustrate the former results.     

Path integral for a neutral spinning particle in interaction with a rotating magnetic field and a scalar potential

In this paper we consider a neutral spinning particle in interaction with a linear increasing rotating magnetic field and a scalar harmonic potential using the path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger’s model. The calculations are carried out explicitly using fermionic exterior current sources. The problem is then reduced to that of a spinning forced harmonic particle whose spin is coupled to exterior derivative current sources. The result of the propagator is given as a series which is exactly summed up by means of the Laplace transformation and the use of some recurrence formula of the oscillator wave functions. The energy spectrum and the corresponding wave functions are also deduced.     

Stationary photon-atom entanglement and flow equation

A novel strategy for quantum-state engineering is presented by employing the method of flow equation. This strategy offers an optimal procedure for generating stationary entangled states of coupled systems of photons and atoms. To illustrate it, the Jaynes-Cummings model as a simple prototype example is analyzed to exhibit how stationary photon-atom entanglement is established. A nonlocal continuous unitary transformation is explicitly constructed and the associated positive operator-valued measures for the photons and atom are obtained. Then, flow of the entanglement entropy is analyzed. A comment is also made on implementing the unitary operation in the method.     

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p²+V(ix)$p^2+V(ix)$ with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.     

The uncertainties in radial position and radial momentum of an electron in the non-relativistic hydrogen-like atom

Similar to the case of a simple harmonic oscillator, an increase in azimuthal quantum number l will result in simultaneous decrease in both the uncertainty in radial position and the uncertainty in radial momentum for the same principal quantum number n in the non-relativistic hydrogen-like atom. Thus, in some cases of hydrogen-like atom and in the case of a simple harmonic oscillator, the more precisely the position is determined, the more precisely the momentum is known in that instant, and vice versa.     

Perturbative symplectic field theory and Wigner function

We study relativistic quantum field theories in phase space, based on representations of the Poincaré group, using the Moyal product. We develop a perturbative theory for quantizing fields, with functional methods in phase space. The two-point function is related to relativistic Wigner functions for bosons and fermions. As an example we analyze the complex scalar field with quartic self-interaction.     

Periodic solutions of a singularly perturbed delay differential equation

A singularly perturbed differential delay equation of the form

(1)     

Exact solution of the Klein-Gordon equation in the presence of a minimal length

We obtain exact solutions of the (1+1)-dimensional Klein-Gordon equation with linear vector and scalar potentials in the presence of a minimal length. Algebraic approach to the problem has also been studied.     

Path integral treatment for a Coulomb system constrained on D-dimensional sphere and hyperboloid

The propagator relating to the evolution of a particle on the D-sphere and the D-pseudosphere, subjected to the Coulomb potential, was reconsidered in the Faddeev-Senjanovic formalism. The mid-point is privileged. The space-time transformations used make it possible to regularize the singularity and to bring back the problem to its dynamical symmetry SU (1, 1).     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.