scholarly journals Quantum Energy of a Particle in a Finite-potential Well Based Upon Golden Metric Tensor

2019 ◽  
pp. 1-9
Author(s):  
I. I. Ewa ◽  
S. X. K. Howusu ◽  
L. W. Lumbi
Keyword(s):  
Quantum Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Particle Energy ◽  
The Other ◽  
Laplacian Operator ◽  
Potential Well ◽  
Metric Tensor ◽  
Quantum Energy

In our previous work titled “Riemannian Quantum Theory of a Particle in a Finite-Potential Well", we constructed the Riemannian Laplacian operator and used it to obtain the Riemannian Schrodinger equation for a particle in a finite-potential well. In this work, we solved the golden Riemannian Schrodinger equation analytically to obtain the particle energy. The solution resulted in two expressions for the energy of a particle in a finite-potential well. One of the expressions is for the odd energy levels while the other is for the even energy levels.

scholarly journals The spinless relativistic kink-like problem

2015 ◽  
Vol 30 (12) ◽  
pp. 1550062 ◽  
Author(s):  
Wolfgang Lucha ◽  
Franz F. Schöberl
Keyword(s):  
Quantum Theory ◽  
Eigenvalue Problem ◽  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Schrodinger Equation ◽  
Bound State ◽  
State Equation ◽  
Hyperbolic Tangent ◽  
Central Potential ◽  
All Solutions

We constrain the possible bound-state solutions of the spinless Salpeter equation (the most obvious semirelativistic generalization of the nonrelativistic Schrödinger equation) with an interaction between the bound-state constituents given by the kink-like potential (a central potential of hyperbolic-tangent form) by formulating a bunch of very elementary boundary conditions to be satisfied by all solutions of the eigenvalue problem posed by a bound-state equation of this type, only to learn that all results produced by a procedure very much liked by some quantum-theory practitioners prove to be in severe conflict with our expectations.

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.

The interaction representation in the quantum theory of fields

1950 ◽  
Vol 201 (1065) ◽  
pp. 284-296 ◽  
Keyword(s):  
Quantum Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Unitary Transformation ◽  
Present Theory ◽  
Point Of View ◽  
Field Equations ◽  
Generalized Schrödinger Equation ◽  
Invariant Interaction ◽  
Theory Of Fields

The interaction representation has recently been introduced into the quantum theory of fields by Tomonaga and Schwinger. Applications of the theory to interacting meson-photon fields have led to apparent difficulties in determining invariant interaction Hamiltonians. Another troublesome feature is the necessity of verifying the integrability conditions of the so-called generalized Schrödinger equation. In the present paper the theory of the interaction representation is presented from a different point of view. It is shown that if two field operators with the same transformation character satisfy two different field equations, there is a unique unitary transformation connecting the field variables on any space-like surface given such a correspondence on one given space-like surface. A differential equation for determining this unique unitary transformation is found which is the analogue of Tomonaga’s generalized Schrödinger equation. This gives directly and simply an invariant interaction Hamiltonian and renders unnecessary the explicit verification of the integrability of the Schrödinger equation, since this is known to have a unique solution. To illustrate the simplification introduced by the present theory, the interaction Hamiltonian for the interacting scalar meson-photon fields is calculated. The result is the same as that obtained by Kanesawa & Tomonaga, but it is obtained by a straightforward calculation without the need to add terms to make the Hamiltonian an invariant.

scholarly journals THE q-DEFORMED SCHRÖDINGER EQUATION OF THE HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE

1994 ◽  
Vol 09 (22) ◽  
pp. 3989-4008 ◽  
Author(s):  
URSULA CAROW-WATAMURA ◽  
SATOSHI WATAMURA
Keyword(s):  
Euclidean Space ◽  
Difference Equation ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Series Representation ◽  
Exponential Functions ◽  
Creation And Annihilation Operators ◽  
The Creation

We consider the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the q analysis. We represent the Schrödinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.

scholarly journals Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

2015 ◽  
Vol 18 (2) ◽  
pp. 321-350 ◽  
Author(s):  
Siwei Duo ◽  
Yanzhi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Euler Method ◽  
Potential Well ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation ◽  
Infinite Potential Well

AbstractIn this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

ARBITRARY l-WAVE SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR THE SCREEN COULOMB POTENTIAL

2013 ◽  
Vol 22 (06) ◽  
pp. 1350036 ◽  
Author(s):  
SHISHAN DONG ◽  
GUO-HUA SUN ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
State Energy ◽  
Energy Levels ◽  
Bound State ◽  
Screen Coulomb Potential ◽  
Normalization Constant ◽  
Angular Momentum State ◽  
Good Agreement

Using improved approximate schemes for centrifugal term and the singular factor 1/r appearing in potential itself, we solve the Schrödinger equation with the screen Coulomb potential for arbitrary angular momentum state l. The bound state energy levels are obtained. A closed form of normalization constant of the wave functions is also found. The numerical results show that our results are in good agreement with those obtained by other methods. The key issue is how to treat two singular points in this quantum system.

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