We report an exact transparent boundary condition (TBC) on the surface of a rectangular cuboid for the three-dimensional (3D) time-dependent Schrödinger equation. It is obtained as a generalization of the well-known TBC for the 1D Schrödinger equation and of the exact TBC in the rectangular domain for the 3D parabolic wave equation, which we reported earlier. Like all other TBCs, it is nonlocal in time domain and relates the boundary transverse derivative of the wave function at any given time to the boundary values of the same wave function at all preceding times. We develop a discretization of this boundary condition for the implicit Crank-Nicolson finite difference scheme. Several numerical experiments demonstrate evolution of the wave function in free space as well as propagation through a number of 3D spherically symmetrical and asymmetrical barriers, and, finally, scattering off an asymmetrical 3D potential. The proposed boundary condition is simple and robust, and can be useful in computational quantum mechanics when an accurate numerical solution of the 3D Schrödinger equation is required.
Resonant Inelastic X-ray Scattering (RIXS) is the one of the most powerful methods for investigation of the electronic structure of materials, specifically of excitations in correlated electron systems. However the potential of the RIXS technique has not been fully exploited because conventional grating spectrometers have not been capable of achieving the extreme resolving powers that RIXS can utilize. State of the art spectrometers in the soft x-ray energy range achieve ~0.25 eV resolution, compared to the energy scales of soft excitations and superconducting gap openings down to a few meV. Development of diffraction gratings with super high resolving power is necessary to solve this problem. In this paper we study the possibilities of fabrication of gratings of resolving power of up to 10 6 for the 0.5 -1.5 KeV energy range. This energy range corresponds to all or most of the useful dipole transitions for elements of interest in most correlated electronic systems, i.e. oxygen K-edge of relevance to all oxides, the transition metal L 2,3 edges, and the M 4,5 edges of the rare earths. Various approaches based on different kinds of diffraction gratings such as deep-etched multilayer gratings, and multilayer coated echelettes are discussed. We also present simulations of diffraction efficiency for such gratings, and investigate the necessary fabrication tolerances.
The extension of the Fresnel integral to tilted objects is studied using both analytical and numerical approaches. Exact solutions of the parabolic wave equation are used for this purpose. The wavefields produced by a beam propagating at an arbitrary angle θ = π/2 relative to the object surface are investigated. The diffraction patterns are simulated for 1 • ≤ θ ≤ 10 • . It is shown that the inverse problem for tilted objects can be reduced to a Fredholm-type integral equation. Both 2D and 3D geometries are considered. The results may be useful as a theoretical framework for the development of coherent reflection imaging of tilted objects and x-ray imaging of submicron footprints of relativistic electron beams for measurement of their size and shape.
In this paper, an exact three-dimensional transparent boundary condition for the parabolic wave equation in a rectangular computational domain is reported. It is a generalization of the well-known two-dimensional Basakov-Popov-Papadakis transparent boundary condition. It relates the boundary transversal derivative of the wave field at any given longitudinal position to the field values at all preceding computational steps. Several examples demonstrate propagation of light along simple structured optical fibers as well as in x-ray guiding structures. The proposed condition is simple and robust and can help to reduce the size of the computational domain considerably.
A relativistically invariant expression for the number of photons in a free classical electromagnetic field through the currents, that created the field, is derived based on the formula for the total energy-momentum of the field. It is demonstrated that it corresponds to the classical limit of the photon number operator known from the quantum electrodynamics. An expression for the total spin moment of the classical electromagnetic field is derived and it is shown that it can be interpreted in terms of spin moments of elementary photons. Similar to the total number of photons the classical spin moment is the limit of the quantum operator of the total spin. It is revealed that the total number of photons as well as the total spin moment are related to each other through a second order tensor, which can be expressed through currents that created the field.
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