Ilya Feranchuk

The operator method of the approximate solution of the Schrödinger equation

Abstract The regular method of the calculation of nonrelativistic hamiltonian eigenfunctions and eigenvalues is developed. It is shown that simple and sufficiently precise expressions for eigenvalues may be found by means of algebraic transformation.

Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method

Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.

Parametric beam instability of relativistic charged particles in a crystal

Abstract A new mechanism of the instability of an electron (positron) beam under conditions of the parametric X-rays is discovered. Equations, which define the amplitude increment of the instability and the starting current, are obtained. It is shown that a considerable part of the particle energy transforms into coherent X-ray energy due to this parametric instability.

Time dependence of X-ray polarizability of a crystal induced by an intense femtosecond X-ray pulse

The time evolution of the electron density and the resulting time dependence of the X-ray polarizability of a crystal irradiated by highly intense XFEL femtosecond pulses is investigated theoretically. Rate equations for bound electrons and the Boltzmann equation for the unbound electron gas are used in calculations.

Covariant description of X-ray diffraction from anisotropically relaxed epitaxial structures

A general theoretical approach to the description of epitaxial layers with essentially different cell parameters and in-plane relaxation anisotropy is presented.

Influence of Surface Roughness on Evaluation of Stress Gradients in Coatings

Roughness influence on the residual stress gradient evaluation in the case of a grazing incidence X-ray diffraction setup is considered. In this geometry the surface roughness changes essentially the X-ray wave fields of the transmitted and diffracted beams inside the coatings and subsurface regions of bulk samples, and thus influences the refractive properties of the investigated sample area. In turn, the change in the refraction index enforces the re-scale of the informational depth and, consequently, the evaluated stress depth profile. The diffracted amplitude from the crystalline grain located beneath the surface is calculated. The surface roughness is shown to contribute into reconstruction of the real stress gradient profile of the coating.

Operator method for nonperturbative calculation of the thermodynamic values in quantum statistics: diatomic molecular gas

Operator method and cumulant expansion are used for nonperturbative calculation of the partition function and the free energy in quantum statistics. It is shown for Boltzmann diatomic molecular gas with some model intermolecular potentials that the zeroth order approximation of the proposed method interpolates the thermodynamic values with rather good accuracy in the entire range of both the Hamiltonian parameters and temperature. The systematic procedure for calculation of the corrections to the zeroth order approximation is also considered.Operator method and cumulant expansion are used for nonperturbative calculation of the partition function and the free energy in quantum statistics. It is shown for Boltzmann diatomic molecular gas with some model intermolecular potentials that the zeroth-order approximation of the proposed method interpolates the thermodynamic values with rather good accuracy in the entire range of both the Hamiltonian parameters and temperature. The systematic procedure for calculation of the corrections to the zeroth-order approximation is also considered.

Physical background for parameters of the quantum Rabi model

We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode, while the second one describes the interaction with field fluctuations. The first part is usually associated with the resonant cavity mode. This division enables us to determine the applicability condition of the single-mode approximation. In addition we identify simple approximate relations for the description of the eigenstates, eigenfunctions and the time evolution of the quantum Rabi model beyond the rotating wave approximation.

Basics of the Operator Method

The majority of approximate methods for the solution of Schrodinger equation (SE) demonstrated in Chap. 1 is not sufficiently universal and theirs applications in the case of system with many degrees of freedom are bound up with serious difficulties. In the following chapters we consider the method which is proposed as an universal procedure for transformation of the perturbation series and its further applications for various quantum systems. We call this technique operator method (OM) in the sense that all calculations are reduced to the algebraic manipulations with the operator matrix elements without solving any differential or integral equations, in zeroth-order approximation as well as in calculating the successive approximations. The results prove that the OM zeroth-order approximation provides an uniformly fitted estimation of the SE eigenvalues and eigenfunctions in the entire range of the Hamiltonian parameters.

Parametric beam instability of the electron bunch in a crystal

Effect of the parametric beam instability (PBI) of the relativistic electron beam in a crystal was analyzed theoretically in1 (see also2). This effect is analogous to self-amplification of spontaneous emission (SASE) mechanism for X-ray Free Electron Laser (XFEL) with an undulator. However, in the case of PBI the transversal modulation of the beam is defined by channeling of the electron in a crystal and the longitudinal modulation arises because of the parametric X-ray radiation mechanism. It is shown in the present paper that the current density J in the electron bunch typical for FEL facility is enough for the beam self-modulation within the X-ray range if the crystal thickness is larger than the crystal absorption length L ≥ Labs. This process could be used for generation of the coherent X-ray pulses if the time τd of the crystal destruction affected by the electron bunch is less than its passing time τd < L=c. The value τd (J) is estimated for the case of the FLASH electron bunch propagating through a Si crystal.