Nonstationary Schrodinger Equation Research Articles

Features of Quantum-Particle Propagation through Porous Structures

The propagation of wave packets of quantum particles through porous structures in thin semi-transparent films is studied. Numerical solution of the nonstationary Schrodinger equation is used in two-dimensional (for slits) and three-dimensional (round holes) cases. The transmission coefficient of quantum particles is compared with that of classical particles with the same density distribution. The vortex motion of the flux density having a hydrodynamic analog is studied. A variant of the “quantum caustic” is considered upon scattering of the particle wave packet at an obstacle.

Probabilistic Representations for Solutions to Initial Boundary Value Problems for Non-Stationary Schrödinger Equation in d-Hyperball

A construction of probabilistic representations for solutions to initial boundary value problem for the non-stationary Schrodinger equation in d-hyperball first obtained by I. Ibragimov, N. Smorodina, and M. Faddeev is extended to multidimensional case. It is proved that the Wiener process in these representations can be replaced by a random walk approximation. The L2-convergence rates are obtained.

Diffraction Limit in Theory of Light Bullets

An averaged variational procedure has been developed to study the propagation of light bullets in optical fibers using the propagation of light pulses in media with Kerr and quadratic nonlinearities as examples. Criteria for the correct choice of trial solutions are discussed. The condition under which the dispersion length is much greater than the length of diffraction spreading is called the diffraction limit. It is shown that, in this limit, the temporal and spatial dynamics of a bullet are independent of each other, while finding the dynamic parameters of a soliton is reduced to finding various solutions of the nonstationary linear Schrodinger equation for an imaginary quantum particle. The efficiency of the proposed approach is demonstrated by the example of finding solutions in the form of optical vortices in a waveguide for nonlinearities of both types.

A Probabilistic Approximation of the Cauchy Problem Solution for the Schrödinger Equation with a Fractional Derivative Operator

We construct two types of probabilistic approximations of the Cauchy problem solution for the nonstationary Schrodinger equation with a symmetric fractional derivative of order α ∈ (1, 2) at the right-hand side. In the first case, we approximate the solution by mathematical expectation of point Poisson field functionals, and in the second case, we approximate the solution by mathematical expectation of functionals of sums of independent random variables having a power asymptotics of a tail distribution.

Динамика измерений координат квантовых нерелятивистских частиц

Background and Objectives: There are a few different ways to change the quantum mechanical wave function. One is the continuous evolution according to the Schrodinger equation; the others are the collapse provided by strong measurements and the weak continuous measurement. The question arises: Can the strong measurement be considered as a dynamical process. Methods: To answer the question, a dynamic model is proposed to simulate numerically the measurement of the coordinates of quantum particles. By measuring the coordinate, we mean the capture of a particle in a narrow box and the excitation of the stationary wave function in it. We consider the amplitude of the probability of transition to a stationary state as a wave function, and the non-stationary Schrodinger equation is solved using this initial function. The transition to a stationary state was modeled as a gated process. Results and Conclusion: It is shown that when measuring coordinates, along with the instant collapse, there are cases when the measurement takes a short but finite time. In particular, the projective measurement of the coordinate of one of the entangled particles takes some time and is accompanied by the correlation in the probability density of the particles coordinates. Moreover, the correlation is lost over time.

Interaction of Ultracold Neutrons with a Neutron Interference Filter Oscillating in Space

The problem of the interaction of ultracold neutrons with a neutron interference filter oscillating in space is investigated. In the problem, the evolution of the wave packet is considered by means of numerically solving the non-stationary Schrodinger equation by splitting the evolution operator. The filter oscillating in space acts as a quantum modulator of the ultracold neutron flux. The resulting spectra of the transmitted and reflected states are obtained depending on the motion parameters of the interferometer.

Scattering of Wave Packets on the Surface of Topological Insulators in the Presence of Potential Barriers with Magnetization

The nonstationary Schrodinger equation is solved numerically by the Cayley method for wave packets that are formed from surface states on the surface of topological insulators and are scattered by a potential barrier, including a barrier with magnetization. The transmission coefficient and spin density distributions are calculated. Expressions are found for the static transmission coefficient through a barrier with the use of the plane-wave approximation and its generalization for wave packets. It is shown that the two-dimensional nature of wave packets leads to noticeable differences in the behavior of the transmission coefficient compared to that in the plane-wave scattering problem. For instance, two-dimensional packets exhibit a significant suppression of Klein tunneling in some energy regions. The results obtained show that the tunneling and spin density of localized wave-packet-type electronic states in structures based on topological insulators can be affected through potential barriers.

Quantum Analogue of Unstable Limit Cycles of a Periodically Perturbed Inverted Oscillator

To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal field. We propose a type of wave function asymptotically satisfying the operator of initial conditions and still admitting analytic integration of the nonstationary Schrodinger equation. The solution demonstrates that for certain perturbation phases determined by the forcing frequency and the initial indeterminacy of the coordinate, the wave-packet center temporarily stabilizes near the potential maximum for approximately two “natural periods” of the oscillator and then moves to infinity with bifurcations in the drift direction. The effect is not masked by packet spreading, because the packet undergoes anomalous narrowing (collapse) to a size of the order of the characteristic length on the above time interval and its unbounded spreading begins only after this.

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrodinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency $$\Omega $$ , amplitude $$F_{0}$$ and phase $$\phi $$ , i.e. the system with the Hamiltonian of $$\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin $$ $$\left( {\Omega t+\phi } \right) $$ . The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables $$\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }$$ and $$\tau =\omega t$$ . The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading $$\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}$$ which decreases first from quite macroscopic values of $$\sigma _{0} =2^{12,\ldots ,25}$$ to minimal one of $$\sim \!(1/\sqrt{2})$$ at times $$\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) $$ and then grows back unlimitedly. For certain phases $$\phi $$ depending on the $$\Omega /\omega $$ ratio and $$n=\log _2\!\sigma _0 $$ , the mass centre of the packet $$\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }$$ delays approximately two natural ‘periods’ $$\sim \!(4\pi /\omega )$$ in the area of the stationary point and then escapes to ‘ $$+$$ ’ or ‘−’ infinity in a bifurcating way. For ‘resonant’ $$\Omega =\omega $$ , the bifurcation phases $$\phi $$ fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic $$\phi ( {\Omega ,n\rightarrow \infty } )$$ obeying the classical formula $$\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega $$ for initial energy $$E = 0$$ in the wide range of $$\Omega =2^{-4},...,2^{7}$$ .

Integrable Time-Dependent Quantum Hamiltonians.

We formulate a set of conditions under which the nonstationary Schrödinger equation with a time-dependent Hamiltonian is exactly solvable analytically. The main requirement is the existence of a non-Abelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.

Influence of the Geometric Dimensions of a Nanosystem on the Main Characteristics of Charge Exchange with a Negative Ion

The problem of charge exchange between a negative hydrogen ion and nanosystems (thin metal films and island films) is considered. Quantum-mechanical calculations show that the effectiveness of charge exchange is determined by the ion-surface distance and the structure of discrete electron energy levels inside the film. In turn, the structure of the discrete levels depends on the film’s geometric dimensions. The characteristics of charge exchange (the effectiveness of the electron transfer, the probability of charge exchange, the propagation velocity, and the average coordinate of the electron’s position inside the film) are calculated via solution of the nonstationary Schrodinger equation. It is shown that these quantities can also be approximated by a simple linear model of electron propagation.

Coupled harmonic oscillators and their quantum entanglement.

A system of two coupled quantum harmonic oscillators with the Hamiltonian H[over ̂]=1/2(1/m_{1}p[over ̂]_{1}^{2}+1/m_{2}p[over ̂]_{2}^{2}+Ax_{1}^{2}+Bx_{2}^{2}+Cx_{1}x_{2}) can be found in many applications of quantum and nonlinear physics, molecular chemistry, and biophysics. The stationary wave function of such a system is known, but its use for the analysis of quantum entanglement is complicated because of the complexity of computing the Schmidt modes. Moreover, there is no exact analytical solution to the nonstationary Schrodinger equation H[over ̂]Ψ=iℏ∂Ψ/∂t and Schmidt modes for such a dynamic system. In this paper we find a solution to the nonstationary Schrodinger equation; we also find in an analytical form a solution to the Schmidt mode for both stationary and dynamic problems. On the basis of the Schmidt modes, the quantum entanglement of the system under consideration is analyzed. It is shown that for certain parameters of the system, quantum entanglement can be very large.

Electron Spectrum of Nonlinear Cold Emission from a Metal under the Action of a Laser Shot

The nonlinear emission of electrons from a metal under the action of a femtosecond moderate-intensity laser pulse (laser shot) has been studied. A theoretical model of the process has been constructed based on the 1D nonstationary Schrodinger equation in the vacuum half-space with given boundary conditions for the electron wavefunction. This equation has been solved using the Laplace transformation. It has been assumed that the states of free electrons in a metal, which are described by the Sommerfeld theory of metals, are insignificantly influenced by the laser field. The energy spectrum of emitted electrons has been obtained, and its dependence on the parameters of the lased shot has been found. The calculated spectrum of nonlinear electron emission from a tungsten nanotip under the action of a 6.5-fs-long laser shot generating a field of 9.26 V/nm agrees with the experimental data.

Scattering of a particle at a crystal-structure atom

The interaction between an atom located inside a crystal structure and a moving external particle is accompanied by changes in its environment, and as a consequence, the atom is additionally affected by it. The force impact of the image of a moving multiply charged ion on the state of an ultrathin carbon film is calculated. The p possibility of forming a pore in the film as a result of ion propagation through the film is shown. The size of the pore turns out to be strictly correlated with that of the ion wave packet incident from free space on the thin-film carbon crystal structure. It is possible to trace the formation of the boundary with a classically inaccessible region called the caustic surface by numerically solving the three-dimensional nonstationary Schrodinger equation in the problem of the collision between a moving multiply charged ion and a carbon atom. On the whole, the performed calculation confirms the previously obtained results.

Об эволюции сходящегося волнового пакета перевёрнутого квантового осциллятора, вынуждаемого однородным гармоническим полем

The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrodinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In “resonant” case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.

Detection of terahertz radiation by resonant tunneling nanoheterostructures

The phenomenon of terahertz radiation detection by resonant tunneling structures (RTSs) has been studied. The calculations of the changes ΔI0 in the direct current (DC) component I0 under the action of an alternating electric field were carried out by the solution of a nonstationary Schrodinger equation with a time-periodic electric field based on the Floquet mode expansion of the wave functions. The dependences of the DC component I0 in resonant tunneling structures on the frequency ν and AC signal amplitude Vac have been built. It is shown that the ΔI0 value in triple-barrier RTSs at resonance frequency hv ≈ Er2–Er1 (Er1 and Er2 are the energies of the size-quantized levels) can exceed a low-frequency value by more than an order of magnitude. The parameters of the structures have been optmized, in order to use them in the terahertz radiation detectors in the anbsence of an external bias. The possibility of tuning the resonance frequency in the terahertz range by changing the DC bias has been shown.

On the distribution of energy of localized solutions of the Schrödinger equation that propagate along symmetric quantum graphs

From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrodinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrodinger operators on different edges have the same structure). For general local self-adjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.

Studying One of the Problems of Wave Filtration

Motions of particles through a permeable barrier are studied from the standpoint of Schrodinger wave dynamics. We consider time evolution of a substance concentration spot in the vicinity of one or two potential barriers. To solve the problem we developed a computer technology for integration of the non-stationary Schrodinger equation. The technology is based on symbol mapping and the matrix exponential method which was applied earlier to solve the classical heat equation. The developed theory is relevant to development of nanotechnology, in particular, to studies on permeability of nanoporous membranes.

Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra

In the paper, we consider the one-dimensional nonstationary Schrodinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.

In Ukrainian

Within dipole approximation and that of effective mass, the wave function of conduction electron in the quasi-uniform high-frequency electromagnetic field is calculated by the help of nonstationary Schrodinger equation. The kinetics of nonequilibrium interacting quasiparticles in semiconductors in the field of intensive laser radiation is constructed using theory of Dirac quantum transitions.