scholarly journals A long sought result: Closed analytical solutions of the Bohr Hamiltonian with the Morse potential

2019 ◽  
Vol 17 ◽  
pp. 15
Author(s):  
D. Bonatsos ◽  
I. Boztosun ◽  
I. Inci
Keyword(s):  
Analytical Solutions ◽  
Morse Potential ◽  
Asymptotic Iteration Method ◽  
Heavy Nuclei ◽  
Energy Eigenvalues ◽  
Angular Momenta ◽  
Medium Mass ◽  
Unstable Case ◽  
Analytical Expressions ◽  
Two Parameter

Closed analytical solutions of the Morse potential for nonzero angular momenta has been an open problem for decades, solved recently by the Asymptotic Iteration Method (AIM) for solving differential equations. Closed analytical expressions have been obtained for the energy eigenvalues and B(E2) rates of the Bohr Hamiltonian in the γ-unstable case, as well as in an exactly separable rotational case with γ ≈ 0, called the exactly separable Morse (ES-M) solution. All medium mass and heavy nuclei with known β1 and γ1 bandheads have been fitted by using the two-parameter γ-unstable solution for transitional nuclei and the three-parameter ES-M for rotational ones. It is shown that bandheads and energy spacings within the bands are well reproduced for more than 50 nuclei in each case. Comparisons to the fits provided by the Davidson and Kratzer potentials, also soluble by the AIM, are made.

scholarly journals Numerical Solution of Schrodinger Equation for Rotating Morse Potential using Matrix Methods with Fourier Sine Basis

2021 ◽  
Author(s):  
Aditi sharma ◽  
O S K Sastri
Keyword(s):  
Schrödinger Equation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Morse Potential ◽  
Asymptotic Iteration Method ◽  
Hamiltonian Matrix ◽  
Model Parameters ◽  
Matrix Mechanics ◽  
Analytical Expressions ◽  
Free Open Source

In this paper, an elegant and easy to implement numerical method using matrix mechanics approach is proposed, to solve the time independent Schrodinger equation (TISE) for Morse potential. It is specifically applied to non-homogeneous diatomic molecule HCl to obtain its rotating-vibrator spectrum. While matrix diagonalization technique is utilised for solving TISE, model parameters for Morse potential are optimized using variational Monte-Carlo (VMC) approach by minimizing χ 2 − value. Thus, validation with experimental vibrational frequencies is completely numerical based with no recourse to analytical solutions. The ro-vibrational spectra of HCl molecule obtained using the optimized parameters through VMC have resulted in least χ 2 − value as compared to those determined using best parameters from multiple regression analysis of analytical expressions. Numerical algorithm for solving the Hamiltonian matrix has been implemented utilizing Free Open Source Software (FOSS) Scilab and simulation results are matching well with those obtained using analytical solutions from Nikiforov-Uvarov (NU) method and asymptotic iteration method (AIM).

scholarly journals Numerical Solution of Schrodinger Equation for Morse Potential using Matrix Methods with Fourier Sine Basis

2020 ◽  
Author(s):  
Aditi sharma
Keyword(s):  
Schrödinger Equation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Morse Potential ◽  
Asymptotic Iteration Method ◽  
Hamiltonian Matrix ◽  
Model Parameters ◽  
Matrix Mechanics ◽  
Analytical Expressions ◽  
Free Open Source

In this paper, an elegant and easy to implement numerical method using matrix mechanics approach is proposed, to solve the time independent Schrodinger equation (TISE) for Morse potential. It is specifically applied to non-homogeneous diatomic molecule HCl to obtain its rotating-vibrator spectrum. While matrix diagonalization technique is utilised for solving TISE, model parameters for Morse potential are optimized using variational Monte-Carlo (VMC) approach by minimizing $\chi^2-$ value. Thus, validation with experimental vibrational frequencies is completely numerical based with no recourse to analytical solutions. The ro-vibrational spectra of HCl molecule obtained using the optimized parameters through VMC have resulted in least $\chi^2-$ value as compared to those determined using best parameters from multiple regression analysis of analytical expressions. Numerical algorithm for solving the Hamiltonian matrix has been implemented utilizing Free Open Source Software (FOSS) Scilab and simulation results are matching well with those obtained using analytical solutions from Nikiforov-Uvarov (NU) method and asymptotic iteration method (AIM).

Nuclear energy excitation by Bohr Hamiltonian for triaxial nuclei using Killingbeck plus Morse potential

2018 ◽  
Vol 27 (09) ◽  
pp. 1850072 ◽  
Author(s):  
B. Tchana Mbadjoun ◽  
J. M. Ema’a Ema’a ◽  
P. Ele Abiama ◽  
G. H. Ben-Bolie ◽  
P. Owono Ateba
Keyword(s):  
Morse Potential ◽  
Nuclear Energy ◽  
Transition Probabilities ◽  
Asymptotic Iteration Method ◽  
Energy Excitation ◽  
Triaxial Deformation ◽  
Energy Eigenvalues ◽  
Present Solution ◽  
Pekeris Approximation ◽  
Exact Separation

This paper proposes an improved potential for the [Formula: see text]-part of the collective Bohr Hamiltonian, namely, a Killingbeck plus Morse potential, while the [Formula: see text]-part is solved for a triaxial deformation close to [Formula: see text]. The Asymptotic Iteration Method is used, involving the Pekeris approximation, to calculate the energy eigenvalues and the eigenfunctions after an exact separation of the Bohr Hamiltonian into its variables is achieved. The results of these calculations are applied for energy spectra of the low-lying states and for corresponding [Formula: see text] quadrupole transition probabilities of the [Formula: see text] isotopes. Moreover, the results of the present solution are compared with those of the well-known [Formula: see text] and [Formula: see text] models.

scholarly journals Relativistic spin-1 particles with position-dependent mass under the Coulomb interaction: Exact analytical solutions of the DKP equation

2013 ◽  
Vol 91 (3) ◽  
pp. 191-197 ◽  
Author(s):  
M.K. Bahar ◽  
F. Yasuk
Keyword(s):  
Coulomb Interaction ◽  
Analytical Solutions ◽  
Iteration Method ◽  
Asymptotic Iteration Method ◽  
Exact Analytical Solutions ◽  
Dkp Equation ◽  
Energy Eigenvalues ◽  
Relativistic Spin ◽  
Dependent Mass ◽  
Position Dependent Mass

The Duffin–Kemmer–Petiau equation with position-dependent mass for relativistic spin-1 particles under equal vector and scalar Coulomb interaction is studied analytically. The energy eigenvalues and corresponding eigenfunctions are obtained using the asymptotic iteration method.

scholarly journals Closed analytical solutions of Bohr Hamiltonian with Manning-Rosen potential model

2015 ◽  
Vol 24 (11) ◽  
pp. 1550089 ◽  
Author(s):  
M. Chabab ◽  
A. Lahbas ◽  
M. Oulne
Keyword(s):  
Potential Model ◽  
Heavy Nuclei ◽  
Axially Symmetric ◽  
Eigenvalues And Eigenfunctions ◽  
Unstable Nuclei ◽  
Unstable Case ◽  
Two Parameters ◽  
Rosen Potential ◽  
Analytical Expressions ◽  
Good Agreement

In the present paper, we have obtained closed analytical expressions for eigenvalues and eigenfunctions of the Bohr Hamiltonian with the Manning–Rosen potential for [Formula: see text]unstable nuclei as well as exactly separable rotational ones with [Formula: see text]. Some heavy nuclei with known [Formula: see text] and [Formula: see text] bandheads have been fitted by using two parameters in the [Formula: see text]unstable case and three parameters in the axially symmetric prolate deformed one. A good agreement with experimental data has been achieved.

Effect of the velocity-dependent potentials on the energy eigenvalues of the Morse potential

Open Physics ◽  
2012 ◽  
Vol 10 (4) ◽  
Author(s):  
Asim Soylu ◽  
Orhan Bayrak ◽  
Ismail Boztosun
Keyword(s):  
Morse Potential ◽  
State Energy ◽  
Bound State ◽  
Quantum States ◽  
Bound State Energy ◽  
Energy Eigenvalues ◽  
Dependent Term ◽  
Oscillator Type ◽  
Dependent Potential ◽  
Pekeris Approximation

AbstractWe investigate the effect of the isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Morse potential for any quantum states. When the velocity-dependent term is used as a constant parameter, ρ(r) = ρ 0, the energy eigenvalues can be obtained analytically by using the Pekeris approximation. When the velocity-dependent term is considered as an harmonic oscillator type, ρ(r) = ρ 0 r 2, we show how to obtain the energy eigenvalues of the Morse potential without any approximation for any n and ℓ quantum states by using numerical calculations. The calculations have been performed for different energy eigenvalues and different numerical values of ρ 0, in order to show the contribution of the velocity-dependent potential on the energy eigenvalues of the Morse potential.

scholarly journals THE ENERGY EIGENVALUES OF THE TWO DIMENSIONAL HYDROGEN ATOM IN A MAGNETIC FIELD

2006 ◽  
Vol 15 (06) ◽  
pp. 1263-1271 ◽  
Author(s):  
A. SOYLU ◽  
O. BAYRAK ◽  
I. BOZTOSUN
Keyword(s):  
Magnetic Field ◽  
Hydrogen Atom ◽  
Iteration Method ◽  
Weak Magnetic Field ◽  
The Other ◽  
Asymptotic Iteration Method ◽  
Iterative Approach ◽  
Two Dimensional ◽  
Energy Eigenvalues ◽  
The Magnetic Field

In this paper, the energy eigenvalues of the two dimensional hydrogen atom are presented for the arbitrary Larmor frequencies by using the asymptotic iteration method. We first show the energy eigenvalues for the case with no magnetic field analytically, and then we obtain the energy eigenvalues for the strong and weak magnetic field cases within an iterative approach for n=2-10 and m=0-1 states for several different arbitrary Larmor frequencies. The effect of the magnetic field on the energy eigenvalues is determined precisely. The results are in excellent agreement with the findings of the other methods and our method works for the cases where the others fail.

Analytical solutions of fractional Schrödinger equation and thermal properties of Morse potential for some diatomic molecules

2021 ◽  
pp. 2150041
Author(s):  
U. S. Okorie ◽  
A. N. Ikot ◽  
G. J. Rampho ◽  
P. O. Amadi ◽  
Hewa Y. Abdullah
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Morse Potential ◽  
State Energy ◽  
Diatomic Molecules ◽  
Bound State ◽  
Bound State Energy ◽  
Fractional Schrödinger Equation ◽  
Energy Eigenvalues ◽  
Fractional Schrodinger Equation

By employing the concept of conformable fractional Nikiforov–Uvarov (NU) method, we solved the fractional Schrödinger equation with the Morse potential in one dimension. The analytical expressions of the bound state energy eigenvalues and eigenfunctions for the Morse potential were obtained. Numerical results for the energies of Morse potential for the selected diatomic molecules were computed for different fractional parameters chosen arbitrarily. Also, the graphical variation of the bound state energy eigenvalues of the Morse potential for hydrogen dimer with vibrational quantum number and the range of the potential were discussed, with regards to the selected fractional parameters. The vibrational partition function and other thermodynamic properties such as vibrational internal energy, vibrational free energy, vibrational entropy and vibrational specific heat capacity were evaluated in terms of temperature. Our results are new and have not been reported in any literature before.

Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

2011 ◽  
Vol 691 ◽  
pp. 487-497 ◽  
Author(s):  
Rabah Mehaddi ◽  
Olivier Vauquelin ◽  
Fabien Candelier
Keyword(s):  
Theoretical Model ◽  
Asymptotic Analysis ◽  
Flow Rate ◽  
Vertical Velocity ◽  
Analytical Solutions ◽  
Boussinesq Approximation ◽  
Singular Case ◽  
Maximal Height ◽  
Analytical Integration ◽  
Analytical Expressions

AbstractThis paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release.

scholarly journals Simple and Accurate Analytical Solutions of the Electrostatically Actuated Curled Beam Problem

2014 ◽  
Author(s):  
Mohammad I. Younis
Keyword(s):  
Analytical Solutions ◽  
Galerkin Approximation ◽  
Single Mode ◽  
Beam Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Analytical Formulas ◽  
Euler Bernoulli Beam ◽  
Analytical Expressions ◽  
Multi Mode

We present analytical solutions of the electrostatically actuated initially deformed cantilever beam problem. We use a continuous Euler-Bernoulli beam model combined with a single-mode Galerkin approximation. We derive simple analytical expressions for two commonly observed deformed beams configurations: the curled and tilted configurations. The derived analytical formulas are validated by comparing their results to experimental data in the literature and numerical results of a multi-mode reduced order model. The derived expressions do not involve any complicated integrals or complex terms and can be conveniently used by designers for quick, yet accurate, estimations. The formulas are found to yield accurate results for most commonly encountered microbeams of initial tip deflections of few microns. For largely deformed beams, we found that these formulas yield less accurate results due to the limitations of the single-mode approximations they are based on. In such cases, multi-mode reduced order models need to be utilized.

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