scholarly journals What is the boundary condition for the radial wave function of the Schrödinger equation?

2011 ◽  
Vol 79 (6) ◽  
pp. 668-671 ◽  
Author(s):  
Anzor A. Khelashvili ◽  
Teimuraz P. Nadareishvili
Keyword(s):  
Boundary Condition ◽  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Radial Wave Function ◽  
Radial Wave

scholarly journals ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION FOR YUKAWA POTENTIAL WITH VARIABLE MASS IN TOROIDAL COORDINATE USING SUPERSYMMETRIC QUANTUM MECHANICS

2020 ◽  
Vol 17 (35) ◽  
pp. 100-108
Author(s):  
Suci FANIANDARI ◽  
A. SUPARMI ◽  
C. CARI
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Yukawa Potential ◽  
Radial Wave Function ◽  
Variable Mass ◽  
Energy Spectra ◽  
Radial Wave ◽  
Infinite Dimensional ◽  
Quantum Numbers

Schrodinger equation on a toroidal coordinate was proposed in theoretical physics to get the information and the behavior of the system of particle. It was solved just recently in case of a charged scalar particle interacting with a uniform magnetic field, a uniform electric field, and a neutral charge constrained to the surface. The methodology used in the referred work was to solve the Schrodinger equation using an approach outlined in the Whittaker-Watson treatise, which deals with an infinite-dimensional eigenvalue problem and specific particular values of the applied field for eigenfunction problem. In contrast, in the quantum mechanical problem, one had an infinite-dimensional generalized eigenvalue problem. This study aimed to obtain the non-relativistic energy eigenvalue and the radial wave function of the Schrodinger equation under the influence of Yukawa potential. The Supersymmetric Quantum Mechanics (SUSY QM) method was used as a basis to tackle the primary objective of this paper to study the problem of a particle with variable mass in toroidal coordinate. The proper super potential was used to deal with the hyperbolic form of effective potential, and the energy spectra were calculated for different quantum numbers, potential depth, and potential parameters. The radial wave function equation for ground and excited state were obtained. The results showed that the increasing value of the quantum numbers caused the energy spectra of the system to increase to the highest value when the quantum number was equal to the potential parameter, which means the most effective energy value was produced, then it was decreased afterward. While the energy value did not depend on the change of the potential parameter. This property could be used to produce this equation as an application of the previous results, the Schrödinger eigenfunction was used as the starting points to solve the other equation in the same geometrical setting and potential.

The generation of rotational bands by deep, diffuse potentials

1988 ◽  
Vol 66 (4) ◽  
pp. 295-301
Author(s):  
A. C. Merchant
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Radial Wave Function ◽  
Perturbation Expansion ◽  
Attractive Potential ◽  
Radial Coordinate ◽  
Quantization Rule ◽  
Radial Wave ◽  
Rotational Bands

Numerical integration of the single-particle Schrödinger equation containing a deep, diffuse, spherically symmetric, attractive, local potential often gives rise to bound-state energy eigenvalues, which lie on almost perfect rotational bands. Each band is characterized by a constant value of (2n + l), where n is the number of interior nodes in the radial wave function and l is the orbital angular-momentum quantum number of a member state. The constants of near-proportionality between the state energies and l(l + 1) for the various bands are considered. An expression for them derived from the Bohr–Sommerfeld quantization rule is found to give numerical results in excellent agreement with those obtained by direct numerical integration of the Schrödinger equation. However, no analytic expressions could be obtained from it, except in the most trivial cases. Also, an approximate expression for these rotational parameters is derived using the Rayleigh–Schrödinger perturbation theory up to fourth order for the case of any potential that may be expanded in ascending even powers of the radial coordinate, r, for which the perturbation expansion converges. This approximation is excellent if (2n + l + 3/2)[Formula: see text], where m is the mass appearing in the Schrödinger equation, V0 is the depth of the attractive potential, and a is a length parameter characterizing its diffuseness. The general results are illustrated with applications to some deep attractive Gaussian potentials of various widths.

scholarly journals ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION IN BISPHERICAL COORDINATE FOR MOBIUS SQUARE PLUS MODIFIED YUKAWA POTENTIAL USING NIKIFOROV UVAROV FUNCTIONAL ANALYSIS (NUFA) METHOD WITH ITS THERMODYNAMICS PROPERTIES FOR DIATOMIC MOLECULES

2021 ◽  
Vol 17 (37) ◽  
pp. 111-134
Author(s):  
Briant Sabathino Harya WIBAWA ◽  
A SUPARMI ◽  
C CARI
Keyword(s):  
Functional Analysis ◽  
Wave Function ◽  
Partition Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Diatomic Molecules ◽  
Radial Part ◽  
Radial Wave ◽  
Vibrational Partition Function ◽  
Bispherical Coordinates

Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.

scholarly journals Bc AND HEAVY MESON SPECTROSCOPY IN THE LOCAL APPROXIMATION OF THE SCHRÖDINGER EQUATION WITH RELATIVISTIC KINEMATICS

2005 ◽  
Vol 20 (17) ◽  
pp. 4035-4054 ◽  
Author(s):  
SAMEER M. IKHDAIR ◽  
RAMAZAN SEVER
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Order Approximation ◽  
Radial Wave Function ◽  
Bound State ◽  
Local Approximation ◽  
Heavy Meson ◽  
Radial Wave ◽  
Relativistic Kinematics ◽  
Hyperfine Splittings

We present bound state masses of the self-conjugate and non-self-conjugate mesons in the context of the Schrödinger equation taking into account the relativistic kinematics and the quark spins. We apply the usual interaction by adding the spin dependent correction. The hyperfine splittings for the 2S charmonium and 1S bottomonium are calculated. The pseudoscalar and vector decay constants of the Bc meson and the unperturbed radial wave function at the origin are also calculated. We have obtained a local equation with a complete relativistic corrections to a class of three attractive static interaction potentials of the general form V(r) = -Ar-β+κrβ+V0, with β = 1, 1/2, 3/4 which can also be decomposed into scalar and vector parts in the form VV(r) = -Ar-β+(1-∊)κrβ and VS(r) = ∊κrβ+V0; where 0≤∊≤1. The energy eigenvalues are carried out up to the third order approximation using the shifted large-N-expansion technique.

Solusi Persamaan Schrödinger Berbasis Panjang Minimal untuk Potensial Coulomb Termodifikasi

2018 ◽  
Vol 2 (2) ◽  
pp. 43-47
Author(s):  
A. Suparmi, C. Cari, Ina Nurhidayati
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Energy Spectra ◽  
Atomic Particle ◽  
Approximate Analytical Solutions ◽  
Radial Schrödinger Equation

Abstrak – Persamaan Schrödinger adalah salah satu topik penelitian yang yang paling sering diteliti dalam mekanika kuantum. Pada jurnal ini persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Fungsi gelombang dan spektrum energi yang dihasilkan menunjukkan kharakteristik atau tingkah laku dari partikel sub atom. Dengan menggunakan metode pendekatan hipergeometri, diperoleh solusi analitis untuk bagian radial persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Hasil yang diperoleh menunjukkan terjadi peningkatan energi yang sebanding dengan meningkatnya parameter panjang minimal dan parameter potensial Coulomb Termodifikasi. Kata kunci: persamaan Schrödinger, panjang minimal, fungsi gelombang, energi, potensial Coulomb Termodifikasi Abstract – The Schrödinger equation is the most popular topic research at quantum mechanics. The  Schrödinger equation based on the concept of minimal length formalism has been obtained for modified Coulomb potential. The wave function and energy spectra were used to describe the characteristic of sub-atomic particle. By using hypergeometry method, we obtained the approximate analytical solutions of the radial Schrödinger equation based on the concept of minimal length formalism for the modified Coulomb potential. The wave function and energy spectra was solved. The result showed that the value of energy increased by the increasing both of minimal length parameter and the potential parameter. Key words: Schrödinger equation, minimal length formalism (MLF), wave function, energy spectra, Modified Coulomb potential

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Gao-Feng Wei ◽  
Wen-Chao Qiang ◽  
Wen-Li Chen
Keyword(s):  
Schrödinger Equation ◽  
Diatomic Molecule ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Energy Levels ◽  
Bound State ◽  
Radial Wave ◽  
Continuous States ◽  
Analytical Properties ◽  
Function Potential

AbstractThe continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.

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