scholarly journals Quantization of the charge in Coulomb plus harmonic potential

2020 ◽  
Author(s):  
Yoon-Seok Choun ◽  
Sang-Jin Sin
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
General Feature ◽  
Degrees Of Freedom ◽  
Harmonic Potential ◽  
Quantum Numbers ◽  
State Dependent ◽  
Heun’S Equation

Abstract We consider two models where the wave equation can be reduced to the effective Schrödinger equation whose potential contains both harmonic and the Coulomb terms, ω2r2 – a/r. The equation reduces to the biconfluent Heun’s equation, and we find that the charge as well as the energy must be quantized and state dependent. We also find that two quantum numbers are necessary to count radial degrees of freedom and suggest that this is a general feature of differential equation with higher singularity like the Heun’s equation.

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves

2019 ◽  
Vol 33 (24) ◽  
pp. 1950285
Author(s):  
Saviour Worlanyo Akuamoah ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Wave Equation ◽  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Momentum Operator ◽  
Matter Wave ◽  
Relativistic Wave Equation ◽  
Relativistic Energy ◽  
Substitution Method ◽  
Energy And Momentum

In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.

COLLECTIVE SPIN BY LINEARIZATION OF THE SCHRÖDINGER EQUATION FOR NUCLEAR COLLECTIVE MOTION

1988 ◽  
Vol 03 (09) ◽  
pp. 859-866 ◽  
Author(s):  
MARTIN GREINER ◽  
WERNER SCHEID ◽  
RICHARD HERRMANN
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
Collective Motion ◽  
First Order ◽  
Energy And Momentum

The free Schrödinger equation for multipole degrees of freedom is linearized so that energy and momentum operators appear only in first order. As an example, we demonstrate the linearization procedure for quadrupole degrees of freedom. The wave function solving this equation carries a spin. We derive the operator of the collective spin and its eigenvalues depending on multipolarity.

I.—The Molecular Spectra of the Hydrogen Isotopes. I.—Application of the Rotating Vibrator Model to the States of D2

1940 ◽  
Vol 59 ◽  
pp. 1-14
Author(s):  
Ian Sandeman
Keyword(s):  
Power Series ◽  
Schrödinger Equation ◽  
Spectral Data ◽  
Diatomic Molecule ◽  
Arbitrary Function ◽  
Schrodinger Equation ◽  
Molecular Spectra ◽  
Essential Step ◽  
Quantum Numbers ◽  
Nuclear Separation

The theory of the rotating vibrator has been developed by the late J. L. Dunham (1932). The essential step in Dunham's treatment of this question is his replacement of the potential expression occurring in the Schrödinger equation for the diatomic molecule by an arbitrary function in terms of the nuclear separation. When this replacement is made, the Schrödinger equation can be solved by methods developed by Wentzel (1926), Brillouin (1926), and Kramers (1926), and the energy of the rotating vibrator can be expressed as a power series in the quantum numbers in a form convenient for application to spectral data.

The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

2005 ◽  
Vol 37 (1) ◽  
pp. 189-199 ◽  
Author(s):  
Naoufel Ben Abdallah ◽  
Florian Méhats ◽  
Christian Schmeiser ◽  
Rada M. Weishäupl
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Harmonic Potential ◽  
Nonlinear Schrödinger
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