Wave Function Properties and Shifted Energy Levels of Bound States in a Plasma

1991 ◽  
Vol 46 (7) ◽  
pp. 583-589 ◽  
Author(s):  
H. Lehmann ◽  
W. Ebeling
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Levels ◽  
Plasma Composition ◽  
Energy Eigenvalues ◽  
Electron Positron ◽  
New Energy ◽  
Variation Procedure

On the basis of earlier work we show a simple way to estimate the properties of bound states in a plasma. The Bethe-Salpeter equation is approximated by an effective Schrodinger equation. The energy eigenvalues are found via a variation procedure. The treatment is applicated to helium-like bound states and excited hydrogen-like states. The effect of the new energy eigenvalues on the plasma composition is discussed for the symmetrical electron-positron plasma.

scholarly journals SCHRÖDINGER EQUATION FOR HEAVY MESONS EXPANDED IN 1/mQ

1996 ◽  
Vol 11 (03) ◽  
pp. 257-266 ◽  
Author(s):  
TAKAYUKI MATSUKI
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Higher Order ◽  
Wave Functions ◽  
Heavy Mesons ◽  
Higher Order Corrections

Operating just once the naive Foldy-Wouthuysen-Tani transformation on the Schrödinger equation for [Formula: see text] bound states described by a Hamiltonian, we systematically develop a perturbation theory in 1/mQ which enables one to solve the Schrödinger equation to obtain masses and wave functions of the bound states in any order of 1/mQ. There also appear negative components of the wave function in our formulation which contribute also to higher order corrections to masses.

scholarly journals A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

2018 ◽  
Vol 73 (5) ◽  
pp. 407-414 ◽  
Author(s):  
Tigran A. Ishkhanyan ◽  
Vladimir P. Krainov ◽  
Artur M. Ishkhanyan
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Energy Levels ◽  
Hermite Functions ◽  
Exact Equation ◽  
Square Root ◽  
Centrifugal Barrier ◽  
Confluent Hypergeometric Functions

AbstractWe present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x−1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x−2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

The bound state solutions of the D-dimensional Schrödinger equation for the Hulthén potential within SUSY quantum mechanics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750028 ◽  
Author(s):  
H. I. Ahmadov ◽  
M. V. Qocayeva ◽  
N. Sh. Huseynova
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Energy Levels ◽  
Wave Functions ◽  
Hulthén Potential ◽  
Radial Wave ◽  
Energy Eigenvalues ◽  
Hulthen Potential

In this paper, the analytical solutions of the [Formula: see text]-dimensional hyper-radial Schrödinger equation are studied in great detail for the Hulthén potential. Within the framework, a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any [Formula: see text] orbital angular momentum case within the context of the Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transforming each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary [Formula: see text] states for [Formula: see text]-dimensional space.

scholarly journals Bound states of purely relativistic nature

2019 ◽  
Vol 204 ◽  
pp. 01014 ◽  
Author(s):  
V.A. Karmanov ◽  
J. Carbonell ◽  
H. Sazdjian
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Levels ◽  
Massive Particle ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Balmer Series ◽  
Photon Exchange ◽  
Exchange Form

Two particles interacting by photon exchange, form the bound states predicted by the non-relativistic Schrödinger equation with the Coulomb potential (Balmer series). More than 60 years ago, in the solutions of relativistic Bethe-Salpeter equation, in addition to the Balmer series, were found another series of energy levels. These new series, appearing when the fine structure constant α is large enough (α > π/4), are not predicted by the Schrödinger equation. However, this new (non-Balmer) states can hardly exist in nature, since in order to create a strong e.m. field with α > π/4 a point-like charge Z > 107 is needed. The nuclei having this charge, though exist starting with bohrium, are far from to be point-like. In the present paper, we analyze the more realistic case of a strong interaction created by exchange of a massive particle. It turns out that in the framework of the Bethe-Salpeter equation this interaction still generates a series of new relativistic states, which are similar to those of the massless exchange case, and which are absent in the Schrödinger equation. The properties of these solutions are studied. Their existence in nature seems possible.

scholarly journals Solutions of the D-Dimensional Schrödinger Equation with the Hyperbolic Pöschl-Teller Potential plus Modified Ring-Shaped Term

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Ibsal A. Assi ◽  
Akpan N. Ikot ◽  
E. O. Chukwuocha
Keyword(s):  
Wave Function ◽  
Energy Level ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Radial Equation ◽  
Energy Eigenvalues ◽  
Separation Of Variable ◽  
Nu Method ◽  
Hyperspherical Coordinate

We solve the D-dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well.

scholarly journals SOLVING THE SCHRÖDINGER EQUATION FOR BOUND STATES WITH MATHEMATICA 3.0

1999 ◽  
Vol 10 (04) ◽  
pp. 607-619 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Interaction Potential ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Eigenvalue ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Method Of Solution ◽  
Numerical Integration Procedure

Using Mathematica 3.0, the Schrödinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from [email protected].

The investigation of Carnot engine in the presence of deformed formalism

2021 ◽  
Vol 36 (35) ◽  
Author(s):  
H. Naseri Karimvand ◽  
B. Lari ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Thermodynamic Properties ◽  
Schrödinger Equation ◽  
Single Particle ◽  
Schrodinger Equation ◽  
Carnot Cycle ◽  
Energy Eigenvalues ◽  
Work Done

In this paper, after introducing a kind of [Formula: see text]-deformation in quantum mechanics, first [Formula: see text]-deformed form of Schrödinger equation for a single particle in a box is derived. Then, the energy eigenvalues and wave function in Schrödinger equation are studied. Also, we discuss the Carnot cycle by using of the energy eigenvalues. We obtain the thermodynamic properties such as force, heat transferred, work done and efficiency in the cycle. Finally, all results have satisfied what we had expected before.

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Sameer Ikhdair ◽  
Ramazan Sever
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Jacobi Polynomials ◽  
Wave Functions ◽  
Three Dimensions ◽  
Eigenvalues And Eigenfunctions ◽  
Energy Eigenvalues ◽  
Pseudoharmonic Potential ◽  
Uvarov Method

AbstractA new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $$ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} $$. The energy eigenvalues and eigenfunctions of the bound-states for the Schrödinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.

INFLUENCE OF EXTERNAL FIELDS ON THE KILLINGBECK POTENTIAL: QUASI EXACT SOLUTION

2013 ◽  
Vol 27 (24) ◽  
pp. 1350176 ◽  
Author(s):  
M. HAMZAVI ◽  
S. M. IKHDAIR
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Particle Physics ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
External Fields ◽  
Interacting Particles ◽  
Ansatz Method ◽  
Cornell Potential ◽  
Aharonov Bohm

The Killingbeck potential consists of oscillator potential plus Cornell potential, i.e. ar2+ br - c/r, that it has received a great deal of attention in particle physics. In this paper, we study the energy levels and wave function for arbitrary m-state in two-dimensional (2D) Schrödinger equation (SE) with a Killingbeck potential under the influence of strong external uniform magnetic and Aharonov–Bohm (AB) flux fields perpendicular to the plane where the interacting particles are confined. We use the wave function ansatz method to solve the radial problem of the Schrödinger equation with Killingbeck potential. We obtain the energy levels in the absence of external fields and also find the energy levels of the familiar Coulomb and harmonic oscillator potentials.

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

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