QUASICLASSICAL ANALYSIS OF THREE-DIMENSIONAL SCHRÖDINGER'S EQUATION AND ITS SOLUTION

Three-dimensional Schrödinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the classical Hamilton–Jacobi equation. Each one-dimensional equation obtained after separation is solved by the conventional WKB method. Quasiclassical solution of the angular equation results in the integral of motion [Formula: see text] and the existence of nontrivial solution for the angular quantum number l = 0. Generalization of the WKB method for multi-turning-point problems is given. Exact eigenvalues for solvable and some "insoluble" spherically symmetric potentials are obtained. Quasiclassical eigenfunctions are written in terms of elementary functions in the form of a standing wave.

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QUANTUM FLUCTUATIONS OF THE ANGULAR MOMENTUM AND ENERGY OF THE GROUND STATE

Modern Physics Letters A ◽

10.1142/s0217732398000061 ◽

1998 ◽

Vol 13 (01) ◽

pp. 33-37 ◽

Cited By ~ 4

Author(s):

M. N. SERGEENKO

Keyword(s):

Angular Momentum ◽

Ground State ◽

Three Dimensional ◽

Quantum Fluctuations ◽

Schrödinger’S Equation ◽

Schrödinger's Equation

Quasiclassical solution of the three-dimensional Schrödinger's equation is given. It is shown apparently that the exitence of nonzero minimal angular momentum M0=ℏ/2 corresponds to the quantum fluctuations of the angular momentum and contributes to the energy of the ground state.

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Pushing the limit for the grid-based treatment of Schrödinger's equation: a sparse Numerov approach for one, two and three dimensional quantum problems

Physical Chemistry Chemical Physics ◽

10.1039/c6cp06698d ◽

2016 ◽

Vol 18 (46) ◽

pp. 31521-31533 ◽

Cited By ~ 8

Author(s):

Ulrich Kuenzer ◽

Jan-Andrè Sorarù ◽

Thomas S. Hofer

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Three Dimensional ◽

Higher Dimensions ◽

Second Order ◽

Special Focus ◽

Schrödinger’S Equation ◽

Numerov Method ◽

Schrödinger's Equation ◽

Grid Based

The general Numerov method employed to numerically solve ordinary differential equations of second order was adapted with a special focus on matrix sparsity and applications in higher dimensions.

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Three-dimensional Spectral Solution of Schrödinger Equation

VLSI Design ◽

10.1155/2001/76808 ◽

2001 ◽

Vol 13 (1-4) ◽

pp. 341-347 ◽

Cited By ~ 2

Author(s):

A. Trellakis ◽

U. Ravaioli

Keyword(s):

Eigenvalue Problem ◽

Conduction Band ◽

Krylov Subspace ◽

Fourier Transforms ◽

Three Dimensional ◽

High Energy ◽

Oxide Semiconductor ◽

Schrödinger’S Equation ◽

Spectral Solution ◽

Schrödinger's Equation

We present a fast and robust method for the full-band solution of Schrödinger's equation on a grid, with the goal of achieving a more complete description of high energy states and realistic temperatures. Using Fast Fourier Transforms, Schrödinger's equation in the one band approximation can be expressed as an iterative eigenvalue problem for arbitrary shapes of the conduction band. The resulting eigenvalue problem can then be solved using Krylov subspace methods as Arnoldi iteration. We demonstrate the algorithm by presenting an example concerning non-parabolic effects in an ultra-small Metal-Oxide-Semiconductor quantum cavity at room-temperature. For this structure, we show that the non-parabolicity of the conduction band results in a significant lowering of high-energy electronic states.

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Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2020-23-3-306-311 ◽

2020 ◽

Vol 23 (3) ◽

pp. 306-311

Author(s):

Yu. Kurochkin ◽

Dz. Shoukavy ◽

I. Boyarina

Keyword(s):

Constant Curvature ◽

Jacobi Equation ◽

Separation Of Variables ◽

Center Of Mass ◽

Three Dimensional ◽

Relative Distance ◽

Dimensional Sphere ◽

Body System ◽

Spaces Of Constant Curvature ◽

Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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