New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

1984 ◽  
Vol 5 (2) ◽  
pp. 1151-1162
Author(s):  
Dong Ming-de
Keyword(s):  
Helmholtz Equation ◽  
Wave Functions ◽  
New Method

New Method for Calculating Wave Functions in Crystals and Molecules

1959 ◽  
Vol 116 (2) ◽  
pp. 287-294 ◽  
Author(s):  
James C. Phillips ◽  
Leonard Kleinman
Keyword(s):  
Wave Functions ◽  
New Method

scholarly journals Relation between Schrödinger’s Equation and Einstein’s Equation

2021 ◽  
Author(s):  
Aman Yadav
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Helmholtz Equation ◽  
Space Time ◽  
Wave Functions ◽  
Quantum Level ◽  
Schrödinger’S Equation ◽  
Einstein's Equation ◽  
The Relationship ◽  
Schrödinger's Equation

The relationship between Einstein's Field Equation and Schrodinger's Equation is examined in thiswork. I adjusted Schrodinger's Equation to offer the solution, and utilizing the wave equation, Icame up with two cases: In case 1, I discovered the structure and dimension of the equations in amanner similar to Einstein's Field Equation, and in case 2, the Helmholtz equation replaces themodified Schrodinger's equation. Finally, the findings suggested that wave functions may haverelevance beyond determining the position of a particle, and that they may be used to determinethe structure of space-time at the quantum level.

A New Method for Calculating Wave Functions in Crystals

1940 ◽  
Vol 57 (12) ◽  
pp. 1169-1177 ◽  
Author(s):  
Conyers Herring
Keyword(s):  
Wave Functions ◽  
New Method

scholarly journals AN APPROXIMATION PROPERTY OF IMPORTANCE IN INVERSE SCATTERING THEORY

2001 ◽  
Vol 44 (3) ◽  
pp. 449-454 ◽  
Author(s):  
David Colton ◽  
Brian D. Sleeman
Keyword(s):  
Helmholtz Equation ◽  
Scattering Theory ◽  
Sampling Method ◽  
Approximation Property ◽  
Wave Functions ◽  
Field Pattern ◽  
Primary 35R30 ◽  
Linear Sampling Method ◽  
Secondary 35P25 ◽  
Far Field Pattern

AbstractA key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle $D$ from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in $D$ can be approximated in $H^1(D)$ by Herglotz wave functions.To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space $H^1(D)$ norm.AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25

scholarly journals The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

2020 ◽  
pp. 1-22
Author(s):  
J. A. Barceló ◽  
M. Folch-Gabayet ◽  
T. Luque ◽  
S. Pérez-Esteva ◽  
M. C. Vilela
Keyword(s):  
Helmholtz Equation ◽  
Sobolev Spaces ◽  
Fractional Integration ◽  
Extension Operator ◽  
Wave Functions ◽  
Square Function ◽  
Entire Solutions ◽  
Fourier Extension ◽  
Fractional Integration Operator ◽  
Weighted Norms

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝ d ) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

scholarly journals The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

2021 ◽  
Author(s):  
◽  
Rufus M Boyack
Keyword(s):  
Helmholtz Equation ◽  
Hydrogen Molecule ◽  
Separation Of Variables ◽  
New Method ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Prolate Spheroids ◽  
Heun Functions ◽  
Angular Equation ◽  
Spheroidal Coordinates

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

scholarly journals Nonadiabatic rotational states of the hydrogen molecule

2018 ◽  
Vol 20 (1) ◽  
pp. 247-255 ◽  
Author(s):  
Krzysztof Pachucki ◽  
Jacek Komasa
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hydrogen Molecule ◽  
High Accuracy ◽  
Wave Functions ◽  
New Method ◽  
Body System ◽  
Coulomb Interactions ◽  
Rotational States

A new method of solving the Schrödinger equation to a high accuracy for a four-body system with Coulomb interactions using exponential wave functions.

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