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The Symposium on Schrodinger Equation of Harmonic Oscillator

Qi Mou

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Symposium is a new calculation method on Hamiltonian mechanics. We use it to solve Schrodinger equation of linear harmonic oscillator on quantum mechanics and perform calculation with computer program made by ourselves,and compare with the ordinary calculation method. The result indicates that the symposium is a very advanced calculation method to solve partial differential equations such as Schrodinger equation.

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Symposium is a new calculation method on Hamiltonian mechanics. We use it to solve Schrodinger equation of linear harmonic oscillator on quantum mechanics and perform calculation with computer program made by ourselves,and compare with the ordinary calculation method. The result indicates that the symposium is a very advanced calculation method to solve partial differential equations such as Schrodinger equation.
Laplace transformation has been used for solving Schrödinger equation in momentum space without using integral equation (Fourier transformation) because it is much harder and more complicated. Furthermore, energy spectra and particle distributions have been demonstrated by using Schrödinger differential equation without ignoring dimensions or units. In addition, a maple program has also been considered to solve the method since it can help to pave a way for other models to work in quantum mechanics.
We write a computer program that uses the recursion relation to calculate wave function in the harmonic-oscillator potential for specified values of E/hv (with its deviation 0.001) containing only even numbers of v (0,2,4,...). In this work, differential equations are solved by power series methods, i.e. the Schrodinger equation stationary state wavefunction of the harmonic-oscillator. This technique is applied to obtain the wavefunction of the quantum harmonic oscillator and is at a more sophisticated level than elsewhere in the course of quantum physics. For the able student this can be a worthwhile extension to the work on the harmonic-oscillator.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
The varying regularity of relative errors of wave function is uncover ed while the Schrodinger equation of harmonic oscillator is solved with symposiu m. It has been found through calculation that relative errors of wave function i n different X points are all the same a t the same time. The relative errors of real number part and imaginary number pa rt of wave function change periodically between positive and negative numbers wi th the passage of time. The period is 628 steps or π seconds. The relation betw een the relative error of real number part and the one of imaginary number part of wave function is similar to the characteristic of Uncertainty Principle. Whe n one relative error approaches infinitely small, the another one approaches inf initely great. The product of the two relative errors is a fixed small number. T he absolute value of the small number increases with the passage of time slowly.
A method of solving the Schrödinger equation based on the use of constant particle-particle interaction potential surfaces (IPS) is proposed. The many-body wave function is presented in a configuration interaction form, with coefficients depending on the total interaction potential. The corresponding set of linear ordinary differential equations for the coefficients was developed. To reduce the computational work, a hierarchy of approximations based on interaction potential surfaces of a part of the particle system was worked out. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with a basis set increase.