scholarly journals On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Bernstein Polynomials ◽  
Energy Levels ◽  
Basis Functions ◽  
Spline Collocation ◽  
Stationary Schrödinger Equation ◽  
B Splines ◽  
Domain Length

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

Local Discontinuous Galerkin Methods for the 2D Simulation of Quantum Transport Phenomena on Quantum Directional Coupler

2014 ◽  
Vol 15 (4) ◽  
pp. 1012-1028 ◽  
Author(s):  
Li Guo ◽  
Yan Xu
Keyword(s):  
Schrödinger Equation ◽  
Quantum Transport ◽  
Schrodinger Equation ◽  
Transport Phenomena ◽  
Basis Functions ◽  
Galerkin Methods ◽  
Stationary Schrödinger Equation ◽  
Local Discontinuous Galerkin ◽  
Local Discontinuous Galerkin Methods ◽  
Ldg Method

AbstractIn this paper, we present local discontinuous Galerkin methods (LDG) to simulate an important application of the 2D stationary Schrödinger equation called quantum transport phenomena on a typical quantum directional coupler, which frequency change mainly reflects in y-direction. We present the minimal dissipation LDG (MD-LDG) method with polynomial basis functions for the 2D stationary Schrödinger equation which can describe quantum transport phenomena. We also give the MD-LDG method with polynomial basis functions in x-direction and exponential basis functions in y-direction for the 2D stationary Schrödinger equation to reduce the computational cost. The numerical results are shown to demonstrate the accuracy and capability of these methods.

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.

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