Second-order relativistic corrections for the S(L=0) states in one- and two-electron atomic systems

2005 ◽  
Vol 83 (1) ◽  
pp. 1-21
Author(s):  
Alexei M Frolov ◽  
Catalin C Mitelut ◽  
Zheng Zhong
Keyword(s):  
Second Order ◽  
Electron Systems ◽  
Specific Interest ◽  
Atomic Systems ◽  
Expectation Value ◽  
Relativistic Corrections ◽  
Alternative Approach ◽  
Compensation Technique ◽  
Three Body ◽  
Regular Operators

An analytical approach is developed to compute the first- (~α2) and second-order (~α4) relativistic corrections in one- and two-electron atomic systems. The approach is based on the reduction of all operators to divergent (singular) and nondivergent (regular) parts. Then, we show that all the divergent parts from the differentmatrix elements cancel each other. The remaining expression contains only regular operators and its expectation value can be easily computed. Analysis of the S(L = 0) states in such systems is of specific interest since the corresponding operators for these states contain a large number of singularities. For one-electron systems the computed relativistic corrections coincide exactly with the appropriate result that follows from the Taylor expansion of the relativistic (i.e., Dirac) energy. We also discuss an alternative approach that allows one to cancel all singularities by using the so-called operator-compensation technique. This second approach is found to be very effective in applications of more complex systems, such as helium-like atoms and ions, H+2-like ions, and some exotic three-body systems.

scholarly journals Semiclassical calculations of the quadruple excited four-electron systems

2007 ◽  
pp. 33-44
Author(s):  
N. Simonovic ◽  
M. Predojevic ◽  
V. Pankovic ◽  
P. Grujic
Keyword(s):  
Energy Levels ◽  
Point Of View ◽  
Interstellar Space ◽  
Vibrational Modes ◽  
Electron Systems ◽  
Atomic Systems ◽  
Excited Atoms ◽  
Relative Contribution ◽  
Quantum Numbers ◽  
Semiclassical States

Highly excited atoms acquire very large dimensions and can be present only in a very rarified gas medium, such as the interstellar space. Multiply excited beryllium-like systems, when excited to large principal quantum numbers, have a radius of r ? 10 ?. We examine the semiclassical spectrum of quadruple highly excited four-electron atomic systems for the plane model of equivalent electrons. The energy of the system consists of rotational and vibrational modes within the almost circular orbit approximation, as used in a previous calculation for the triply excited three-electron systems. Here we present numerical results for the beryllium atom. The lifetimes of the semiclassical states are estimated via the corresponding Lyapunov exponents. The vibrational modes relative contribution to the energy levels rises with the degree of the Coulombic excitation. The relevance of the results is discussed both from the observational and heuristic point of view.

scholarly journals A Hida–Malliavin white noise calculus approach to optimal control

2018 ◽  
Vol 21 (03) ◽  
pp. 1850014
Author(s):  
Nacira Agram ◽  
Bernt Øksendal
Keyword(s):  
Optimal Control ◽  
Diffusion Coefficient ◽  
White Noise ◽  
Backward Stochastic Differential Equation ◽  
Second Order ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Alternative Approach ◽  
Quadratic Optimal Control ◽  
White Noise Calculus

The classical maximum principle for optimal stochastic control states that if a control [Formula: see text] is optimal, then the corresponding Hamiltonian has a maximum at [Formula: see text]. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

scholarly journals First order normalization in the generalized photo gravitational non-planar restricted three body problems

2015 ◽  
Vol 3 (2) ◽  
pp. 46
Author(s):  
Nirbhay Kumar Sinha
Keyword(s):  
Oblate Spheroid ◽  
Second Order ◽  
Elliptic Motion ◽  
First Order ◽  
Short Period ◽  
Order Part ◽  
Three Body

<p>In this paper, we normalised the second-order part of the Hamiltonian of the problem. The problem is generalised in the sense that fewer massive primary is supposed to be an oblate spheroid. By photogravitational we mean that both primaries are radiating. With the help of Mathematica, H<sub>2</sub> is normalised to H<sub>2</sub> = a<sub>1</sub>b<sub>1</sub>w<sub>1</sub> + a<sub>2</sub>b<sub>2</sub>w<sub>2</sub>. The resulting motion is composed of elliptic motion with a short period (2p/w<sub>1</sub>), completed by an oscillation along the z-axis with a short period (2p/w<sub>2</sub>).</p>

Temperature Dependence of Second and Higher Order Elastic Constants of Orientationally Disordered (NH4I)x (KI)1-x Mixed Crystals

2014 ◽  
Vol 1047 ◽  
pp. 65-70 ◽  
Author(s):  
Alpana Tiwari
Keyword(s):  
Elastic Constants ◽  
Body Force ◽  
Coupling Effect ◽  
Second Order ◽  
Mixed Crystals ◽  
Elastic Behaviour ◽  
Coupling Term ◽  
Third Order ◽  
Third Order Elastic Constants ◽  
Three Body

We have incorporated the translational rotational (TR) coupling effects in the framework of three body force shell model (TSM) to develop an extended TSM (ETSM). This ETSM has been applied to reveal the second order elastic constants (C11, C12and C44) in the dilute regimes 0≤ x ≤ 0.50 as a function of temperature for 10K≤T≤300K. The anomalous elastic behaviour in C44below 100 K has been depicted well by ETSM results in the orientationally disordered (NH4I)x(KI)1-xmixed crystals. In order to present a visual comparison of the TR-coupling effect on second order elastic constants, we have evaluated the SOECs with and without TR coupling term in ETSM. Besides third order elastic constants have also been studied and discussed for concentration range 0≤x≤0.50 as a function of temperature for 10K≤T≤300K.

The Changing Views of a Philosopher of Chemistry on the Question of Reduction

2016 ◽  
Author(s):  
Eric R. Scerri
Keyword(s):  
Quantum Mechanics ◽  
Mathematical Theory ◽  
Approximate Solutions ◽  
Electron Systems ◽  
Atomic Systems ◽  
Quantum Mechanical Description ◽  
The Subject ◽  
The Moment ◽  
The Relationship

The question of the reduction of chemistry to quantum mechanics has been inextricably linked with the development of the philosophy of chemistry since the field began to develop in the early 1990s. In the present chapter I would like to describe how my own views on the subject have developed over a period of roughly 30 years. A good place to begin might be the frequently cited reductionist dictum that was penned in 1929 by Paul Dirac, one of the founders of quantum mechanics. . . . The underlying laws necessary for the mathematical theory of a larger part of physics and the whole of chemistry are thus completely known, and the difficulty is only that exact applications of these laws lead to equations, which are too complicated to be soluble. (Dirac 1929) . . . These days most chemists would probably comment that Dirac had things backward. It is clear that nothing like “the whole of chemistry” has been mathematically understood. At the same time most would argue that the approximate solutions that are afforded by modern computers are so good as to overcome the fact that one cannot obtain exact or analytical solutions to the Schrödinger equation for many-electron systems. Be that as it may, Dirac’s famous quotation, coming from one of the creators of quantum mechanics, has convinced many people that chemistry has been more or less completely reduced to quantum mechanics. Another quotation of this sort (and one using more metaphorical language) comes from Walter Heitler who together with Fritz London was the first to give a quantum mechanical description of the chemical bond. . . . Let us assume for the moment that the two atomic systems ↑↑↑↑ . . . and ↓↓↓↓ . . . are always attracted in a homopolar manner. We can, then, eat Chemistry with a spoon. (Heitler 1927) . . . Philosophers of science eventually caught up with this climate of reductionism and chose to illustrate their views with the relationship with chemistry and quantum mechanics.

scholarly journals Ground state for two-electron and electron-muon three-body atomic systems

2009 ◽  
pp. NA-NA ◽  
Author(s):  
K. V. Rodriguez ◽  
L. U. Ancarani ◽  
G. Gasaneo ◽  
D. M. Mitnik
Keyword(s):  
Ground State ◽  
Atomic Systems ◽  
Three Body

Retardation in non-dispersive interactions between molecules

1966 ◽  
Vol 295 (1443) ◽  
pp. 476-489 ◽  
Keyword(s):  
Quantum Electrodynamics ◽  
Exchange Interactions ◽  
Dipole Interaction ◽  
Second Order ◽  
The Other ◽  
Spin Orbit ◽  
Wave Zone ◽  
Expectation Value ◽  
Near Zone ◽  
The Matrix

The second order T matrix corresponding to the interaction between two molecules is calculated by quantum electrodynamics. In the near zone the matrix reduces to the expectation value of the Breit Hamiltonian for the two-centre problem. In the wave zone a retarded Briet operator is found for exchange interactions. A reduction to the Pauli limit is made. The interactions are discussed severally for the spin-spin, (spin-dipole)-(spin-dipole), spin-orbit and dipole-(spin-dipole) cases. At large separations the T matrix is complex and the imaginary parts, previously given for the dipole-dipole interaction, are found for the other cases.

scholarly journals A Harmonic Voltage Injection Based DC-Link Imbalance Compensation Technique for Single-Phase Three-Level Neutral-Point-Clamped (NPC) Inverters

Energies ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1886 ◽  
Author(s):  
Kyoung-Pil Kang ◽  
Younghoon Cho ◽  
Myung-Hyo Ryu ◽  
Ju-Won Baek
Keyword(s):  
Second Order ◽  
Neutral Point ◽  
Voltage Difference ◽  
Imbalance Problem ◽  
Offset Voltage ◽  
Half Wave ◽  
Unbalanced Voltage ◽  
Compensation Technique ◽  
Harmonic Voltage ◽  
Harmonic Components

In three-level neutral-point-clamped (NPC) inverters, the voltage imbalance problem between the upper and lower dc-link capacitors is one of the major concerns. This paper proposed a dc-link capacitor voltage balancing method where a common offset voltage was injected. The offset voltage consists of harmonic components and a voltage difference between the upper and the lower capacitors. Here, both the second-order harmonics and the half-wave of the second-order component were injected to compensate for the unbalanced voltage between the capacitors. In order to show the effectiveness of the proposed voltage injection, the theoretical analyses, simulations, and experimental results are provided. Since the proposed method does not require any hardware modifications, it can be easily adapted. Both the simulations and the experiments validated that the voltage difference of the dc-link could be effectively reduced with the proposed method.

Sign in / Sign up

Export Citation Format

Share Document