Schrödinger’s Equation with Potential Energy: Introduction to Operators

2017 ◽  
pp. 69-76
Author(s):  
Paul R. Berman
Keyword(s):  
Potential Energy ◽  
Schrödinger’S Equation ◽  
Schrödinger's Equation

A Theory of Hartree's Atomic Fields

1928 ◽  
Vol 24 (2) ◽  
pp. 328-342 ◽  
Author(s):  
J. A. Gaunt
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Potential Energy ◽  
Unit Volume ◽  
Wave Functions ◽  
The Other ◽  
Schrödinger’S Equation ◽  
Energy Parameters ◽  
Old Quantum Theory ◽  
Schrödinger's Equation

In a recent communication to this Society Dr Hartree has put forward a method for calculating the field of anatom containing many electrons. Each orbit—to borrow a metaphor from the old quantum theory—is related to a wave-function Ψ which obeys Schrödinger's equation. The potential energy used in this equation is due partly to the field of the nucleus, and partly to the fields of the electrons in the other orbits. The latter are calculated upon Schrödinger's interpretation of the wave-function, that |Ψ|2 is the density of charge, measured in electronic charges per unit volume. It is not the purpose of this paper to discuss the practical methods of obtaining wave-functions which reproduce the fields from which they are derived; but to relate these wave-functions and their energy parameters to those of the accepted theory.

scholarly journals Orthogonal structure on a quadratic curve

2020 ◽  
Author(s):  
Sheehan Olver ◽  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Extension Problem ◽  
Orthogonal Expansions ◽  
Schrödinger’S Equation ◽  
Fourier Extension ◽  
Quadratic Curve ◽  
Interpolation Of Functions ◽  
Fourier Orthogonal Expansions ◽  
Schrödinger's Equation

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

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