scholarly journals The relativistic feature of Hydrogen-like atoms in Heisenberg picture

2021 ◽  
Author(s):  
Jafar Jahanpanah ◽  
A Vahedi ◽  
H . Khosrojerdi
Keyword(s):  
Energy Levels ◽  
Energy Eigenvalue ◽  
Exact Expression ◽  
Linear Momentum ◽  
Heisenberg Equation ◽  
Relativistic Energy ◽  
Heisenberg Picture ◽  
First Order ◽  
Coupled Equations ◽  
Atomic Energy Levels

The relativistic behavior of Hydrogen-like atoms (HLAs) is investigated in Heisenberg picture for the first time. The relativistic vibrational Hamiltonian (RVH) is first defined as a power series of harmonic oscillator Hamiltonian by using the relativistic energy eigenvalue . By applying the first-order RVH (proportional to ) to Heisenberg equation, a pair of coupled equations is turned out for the motion of electron position and its relativistic linear momentum. A simple comparison of the first-order relativistic and nonrelativistic equations reveals this reality that the natural (fundamental) frequency of electron oscillation (like entropy) is slowly raised by increasing the atomic number. The second-order RVH (proportional to ) have then been implemented to determine an exact expression for the electron relativistic frequency in the different atomic energy levels. In general, the physical role of RVH is fundamental because it not only specifies the temporal relativistic variations of position, velocity, and linear momentum of oscillating electron, but also identifies the corresponding relativistic potential, kinetic, and mechanical energies. The results will finally be testified by demonstrating the energy conservation.

RELATIVISTIC ENERGY, OSCILLATOR STRENGTHS AND TRANSITION RATES OF THE 1s2 2 ln l 1S(m)(n =2–6, m1–5) STATES FOR Be-LIKE SYSTEM

2005 ◽  
Vol 16 (06) ◽  
pp. 951-968 ◽  
Author(s):  
MENG ZHANG ◽  
BING-CONG GOU
Keyword(s):  
Experimental Data ◽  
Energy Levels ◽  
Oscillator Strengths ◽  
Relativistic Energy ◽  
Transition Rates ◽  
Isoelectronic Sequence ◽  
Polarization Effects ◽  
First Order ◽  
Variational Calculations ◽  
First Order Perturbation

Variational calculations are carried out with a multiconfiguration-interaction wave function to obtain the relativistic energies of the 1s2 2 ln l 1 S (m)(n =2–6, m1–5) states for the beryllium isoelectronic sequence (Z =4–10). Relativistic corrections and the mass polarization effects are evaluated with the first-order perturbation theory. The identifications of the energy levels for 1s2 2 ln l 1 S (m)(n =2–6, m1–5) states in the Be-like ions are reported. The oscillator strengths, transition rates and wavelengths are also calculated. The calculated results are compared with other theoretical and experimental data in the literature.

scholarly journals Relativistic Corrections to Atomic Energy Levels

1971 ◽  
Vol 2 ◽  
pp. 549-554
Author(s):  
Michael Jones
Keyword(s):  
Computer Program ◽  
Energy Levels ◽  
Relativistic Effects ◽  
General Purpose ◽  
Wave Functions ◽  
Relativistic Energy ◽  
Intermediate Coupling ◽  
Atomic Systems ◽  
Automatic Computer ◽  
Atomic Energy Levels

AbstractThe relativistic effects on bound levels of atomic systems with particular application to systems of astrophysical importance are investigated by including the most important contributions to the low Z Pauli Approximation for the Hamiltonian. This work is an extension of a general purpose automatic computer program for calculating non-relativistic energy levels of complex atoms, described by Eissner and Nussbaumer (1969). The program should give fairly good intermediate coupling wave functions and it may thus be used as a basis for the calculation of intermediate coupling collision strenghts and oscillator strenghts. Numerical results are displayed for the Sodiumi sequence.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Analytical solution of the Shredinger equation for the linear combination of the Manning-Rosen and the class of Yukawa potential

2021 ◽  
pp. 157-169
Author(s):  
G.A. Bayramova ◽  
Keyword(s):  
Analytical Solution ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Yukawa Potential ◽  
Energy Levels ◽  
Energy Eigenvalue ◽  
Radial Wave ◽  
Rosen Potential ◽  
Analytical Expressions ◽  
Potential Parameters

In the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are very sensitive to the choice of potential parameters.

scholarly journals Intercombination Transitions between Levels X1Σg+ and A 3Πu in C2

1987 ◽  
Vol 120 ◽  
pp. 103-105
Author(s):  
J. Le Bourlot ◽  
E. Roueff
Keyword(s):  
Transition Probabilities ◽  
Energy Levels ◽  
Wave Functions ◽  
Spin Orbit Coupling ◽  
Energy Matrix ◽  
First Order ◽  
Emission Transition ◽  
Intercombination Transition ◽  
The One ◽  
Potential Curves

We present a new calculation of intercombination transition probabilities between levels X1Σg+ and a 3Πu of the C2 molecule. Starting from experimental energy levels, we calculate RKR potential curves using Leroy's Near Dissociation Expansion (NDE) method; these curves give us wave functions for all levels of interest. We then compute the energy matrix for the four lowest states of C2, taking into account Spin-Orbit coupling between a 3Πu and A 1Πu on the one hand and X 1Σ+g and b 3Σg− on the other. First order wave functions are then derived by diagonalization. Einstein emission transition probabilities of the Intercombination lines are finally obtained.

Stochastic resonance for a fractional oscillator with random trichotomous mass and random trichotomous frequency

2017 ◽  
Vol 31 (30) ◽  
pp. 1750231 ◽  
Author(s):  
Lifeng Lin ◽  
Huiqi Wang ◽  
Suchuan Zhong
Keyword(s):  
Stochastic Resonance ◽  
Exact Expression ◽  
Order Moment ◽  
Periodic Signal ◽  
Transformation Technique ◽  
First Order ◽  
Fractional Oscillator ◽  
Trichotomous Noise ◽  
Output Amplitude ◽  
Fractional System

The stochastic resonance (SR) phenomena of a linear fractional oscillator with random trichotomous mass and random trichotomous frequency are investigate in this paper. By using the Shapiro–Loginov formula and the Laplace transformation technique, the exact expression of the first-order moment of the system’s steady response is derived. The numerical results demonstrate that the evolution of the output amplitude is nonmonotonic with frequency of the periodic signal, noise parameters and fractional order. The generalized SR (GSR) phenomena, including single GSR (SGSR) and doubly GSR (DGSR), and trebly GSR (TGSR), are detected in this fractional system. Then, the GSR regions in the [Formula: see text] plane are determined through numerical calculations. In addition, the interaction effect of the multiplicative trichotomous noise and memory can diversify the stochastic multiresonance (SMR) phenomena, and induce reverse-resonance phenomena.

Atomic Energy Levels of Neutral Erbium*

1965 ◽  
Vol 55 (5) ◽  
pp. 471 ◽  
Author(s):  
Louis C. Marquet ◽  
Sumner P. Davis
Keyword(s):  
Energy Levels ◽  
Atomic Energy ◽  
Atomic Energy Levels

scholarly journals Solution of Q-Deformed D-Dimensional Klein-Gordon Equation Kratzer Potential using Hypergeometric Method

2019 ◽  
Vol 9 (2) ◽  
pp. 163
Author(s):  
Suparmi Suparmi ◽  
Dyah Ayu Dianawati ◽  
Cari Cari
Keyword(s):  
Gordon Equation ◽  
Energy Levels ◽  
Spin Symmetry ◽  
Relativistic Energy ◽  
Dimensional Parameter ◽  
Energy Value ◽  
Quantum Numbers ◽  
Klein Gordon ◽  
Dimensional Parameters ◽  
Klein Gordon Equation

The Q-deformed D-dimensional Klein Gordon equation with Kratzer potential is solved by using Hypergeometric method in the case of exact spin symmetry. The linear radial momentum of D-dimensional Klein Gordon equation is disturbed by the presence of the quadratic radial posisiton. The Klein-Gordon D-dimensional equation is reduced to one-dimensional Schrodinger like equation with variable substitution. The solution of the D-dimensional Klein-Gordon equation is determined in the form of a general equation of the Hypergeometry function using the Kratzer potential variable and the quantum deformation variable. From this equation, relativistic energy and wave function are determined. In addition, the relativistic energy equation can be used to calculate numerical energy levels for diatomic particles (CO, NO, O2) using Matlab R2013a software. The results obtained show that the q-deformed quantum parameters, quantum numbers and dimensions affect the value of relativistic energy for zero-pin particles. The value of energy increases with increasing value of quantum number n, q-deformed parameters, and d-dimensional parameters. Of the three parameters, q-deformed parameter is the most dominant to give change in energy value; the increasing q-deformed parameter causes the energy value increases significantly compared to the d-dimensional parameter and quantum numbers n.

Sign in / Sign up

Export Citation Format

Share Document