scholarly journals New possibilities of harmonic oscillator basis application for calculation of the ground state energy of a Coulomb non-identical three-particle system

2017 ◽  
Vol 57 (2) ◽  
Author(s):  
Algirdas Deveikis
Keyword(s):  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Particle System ◽  
Traditional Approach ◽  
Expansion Method ◽  
Particle Systems ◽  
Oscillator Basis ◽  
Oscillator Length

A new harmonic oscillator (HO) expansion method for calculation of the non-relativistic ground state energy of the Coulomb non-identical three-particle systems is presented. The HO expansion basis with different size parameters in the Jacobi coordinates instead of only one unique oscillator length parameter in the traditional treatment is introduced. This method is applied to calculate the ground state energy of a number of Coulomb three-particle systems for up to 28 excitation HO quanta. The obtained results suggest that the HO basis with different size parameters in the Jacobi coordinates could lead to significant increasing of the rate of convergence for the ground state energy than in the traditional approach.

scholarly journals Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

2018 ◽  
Vol 173 ◽  
pp. 02006
Author(s):  
Algirdas Deveikis
Keyword(s):  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Traditional Approach ◽  
Coulomb Systems ◽  
Expansion Method ◽  
Variational Parameter ◽  
Oscillator Representation ◽  
Identical Particles ◽  
Oscillator Basis

The variational method in oscillator representation with individual parameters for each Jacobi coordinate is applied to the non-relativistic calculation of the ground state energy of a number of three-particle Coulomb systems, consisting of two identical particles and a different one. The accuracy and convergence rate of the calculations in the constructed oscillator basis are studied up to a total of 28 oscillator quanta. The results are compared with those of the traditional approach using only one such nonlinear variational parameter. The method with individual parameters for Jacobi coordinates is found to possess a number of advantages as compared to the traditional approach.

Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

On the Uniform Upper Bounds to the Ground State Energy of Quantum System

1997 ◽  
Vol 11 (10) ◽  
pp. 1235-1244
Author(s):  
A. N. Kireev
Keyword(s):  
Variational Principle ◽  
Harmonic Oscillator ◽  
Quantum System ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Natural Generalization ◽  
Upper Bounds ◽  
General Character ◽  
Exact Ground State

We derive a set of improving uniform upper bounds to the ground state energy of a quantum system, which provides a natural generalization of the Ritz variational principle. The bounds have a general character, do not depend on the structure of Hamiltonian of a quantum system and converge to its exact ground state energy. As an illustration of the method proposed, we consider a simple example of the shifted harmonic oscillator.

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