scholarly journals Semiexact Solutions of the Razavy Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Qian Dong ◽  
F. A. Serrano ◽  
Guo-Hua Sun ◽  
Jian Jing ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Energy Levels ◽  
Wave Functions ◽  
Potential Parameter ◽  
Heun Functions

In this work, we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on m are illustrated graphically for a given potential parameter ξ. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We also notice that the energy levels ϵi decrease with the increasing potential parameter m.

Exact solutions of a quartic potential

2019 ◽  
Vol 34 (26) ◽  
pp. 1950208 ◽  
Author(s):  
Qian Dong ◽  
Guo-Hua Sun ◽  
M. Avila Aoki ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Analytical Solutions ◽  
Wave Functions ◽  
Negative Potential ◽  
Potential Parameter ◽  
Real Numbers ◽  
Quartic Potential ◽  
Heun Functions ◽  
Potential Parameters

We find that the analytical solutions to quantum system with a quartic potential [Formula: see text] (arbitrary [Formula: see text] and [Formula: see text] are real numbers) are given by the triconfluent Heun functions [Formula: see text]. The properties of the wave functions, which are strongly relevant for the potential parameters [Formula: see text] and [Formula: see text], are illustrated. It is shown that the wave functions are shrunk to the origin for a given [Formula: see text] when the potential parameter [Formula: see text] increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter [Formula: see text] increases or parameter [Formula: see text] decreases for a given negative potential parameter [Formula: see text]. The minimum value of the double well case ([Formula: see text]) is given by [Formula: see text] at [Formula: see text].

Exact solutions to solitonic profile mass Schrödinger problem with a modified Pöschl–Teller potential

2016 ◽  
Vol 31 (04) ◽  
pp. 1650017 ◽  
Author(s):  
Shishan Dong ◽  
Qin Fang ◽  
B. J. Falaye ◽  
Guo-Hua Sun ◽  
C. Yáñez-Márquez ◽  
...  
Keyword(s):  
Wave Function ◽  
Exact Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Wave Functions ◽  
Numerical Calculations ◽  
Potential Parameter ◽  
Heun Functions ◽  
Single Well

We present exact solutions of solitonic profile mass Schrödinger equation with a modified Pöschl–Teller potential. We find that the solutions can be expressed analytically in terms of confluent Heun functions. However, the energy levels are not analytically obtainable except via numerical calculations. The properties of the wave functions, which depend on the values of potential parameter [Formula: see text] are illustrated graphically. We find that the potential changes from single well to a double well when parameter [Formula: see text] changes from minus to positive. Initially, the crest of wave function for the ground state diminishes gradually with increasing [Formula: see text] and then becomes negative. We notice that the parities of the wave functions for [Formula: see text] also change.

scholarly journals Exact Solutions of the Razavy Cosine Type Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Shishan Dong ◽  
Qian Dong ◽  
Guo-Hua Sun ◽  
S. Femmam ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Energy Levels ◽  
Wave Functions ◽  
Potential Parameter ◽  
Heun Function ◽  
Type Potential ◽  
Confluent Heun Function

We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a.

scholarly journals Exact solutions of the harmonic oscillator plus non-polynomial interaction

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200050
Author(s):  
Qian Dong ◽  
H. Iván García Hernández ◽  
Guo-Hua Sun ◽  
Mohamad Toutounji ◽  
Shi-Hai Dong
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Wave Functions ◽  
Potential Parameter ◽  
Potential Well ◽  
One Dimensional ◽  
Minimum Value ◽  
Heun Functions ◽  
Potential Parameters

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a   x 2  +  b   x 2 /(1 +  c   x 2 ) ( a  > 0, c  > 0) are given by the confluent Heun functions H c ( α , β , γ , δ , η ; z ). The minimum value of the potential well is calculated as V min ( x ) = − ( a + | b | − 2 a   | b | ) / c at x = ± [ ( | b | / a − 1 ) / c ] 1 / 2 (| b | >  a ) for the double-well case ( b  < 0). We illustrate the wave functions through varying the potential parameters a , b , c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c . However, we find that the wave peaks are concave to the origin as the parameter | b | is increased.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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.

Relativistic quantum bouncing particles in a homogeneous gravitational field

2021 ◽  
pp. 2150098
Author(s):  
Ar Rohim ◽  
Kazushige Ueda ◽  
Kazuhiro Yamamoto ◽  
Shih-Yuin Lin
Keyword(s):  
Gravitational Field ◽  
Relativistic Effect ◽  
Relativistic Quantum ◽  
Energy Levels ◽  
Nonrelativistic Limit ◽  
Wave Functions ◽  
Dirac Equations ◽  
Rindler Coordinates ◽  
Klein Gordon ◽  
Homogeneous Gravitational Field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

Semi-empirical Molecular Orbital Energy Levels of the Hexammine and Chloroammine Complexes of Co(IV)

1967 ◽  
Vol 22 (2) ◽  
pp. 170-175 ◽  
Author(s):  
Walter A. Yeranos ◽  
David A. Hasman
Keyword(s):  
Molecular Orbital ◽  
Energy Levels ◽  
Empirical Evaluation ◽  
Wave Functions ◽  
Electronic Transitions ◽  
Orbital Energy ◽  
Molecular Orbital Energy ◽  
The One ◽  
Semi Empirical

Using the recently proposed reciprocal mean for the semi-empirical evaluation of resonance integrals, as well as approximate SCF wave functions for Co3+, the one-electron molecular energy levels of Co (NH3) 3+, Co (NH3) 5Cl2+, and Co (NH3) 4Cl21+ have been redetermined within the WOLFSBERG–HELMHOLZ approximation. The outcome of the study fits remarkably well with the observed electronic transitions in the u.v. spectra of these complexes and prompts different band assignments than previously suggested.

scholarly journals Spin-0 system of DKP equation in the background of a flat class of Gödel-type spacetime

2019 ◽  
Vol 35 (07) ◽  
pp. 2050031 ◽  
Author(s):  
Faizuddin Ahmed ◽  
Hassan Hassanabadi
Keyword(s):  
Energy Levels ◽  
Wave Functions ◽  
Dkp Equation ◽  
Free Particles ◽  
The Individual

In this paper, we investigate the Duffin–Kemmer–Petiau (DKP) equation for spin-0 system of charge-free particles in the background of a flat class of Gödel-type spacetimes, and evaluate the individual energy levels and corresponding wave functions in detail.

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