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THE ANALYTICAL NON-SERIES TIME EVOLUTION SOLUTION OF ONE DIMENSIONAL OSCILLATOR SCHRÖDINGER EQUATION

Wenyu Wang, Xue-Ping Cheng, Qiwei LIANG, Linshuo YANG

Published Aug 1, 2024• 0 citations
Publisher

Abstract

In 1926, E.Schrödinger found an analytical solution for the time evolution of the one-dimensional harmonic oscillator Schrödinger Equation. This solution is not in the form of a summation of energy eigenstate wave functions, which provides a new perspective on the time evolution in quantum mechanics. This paper mainly analyzes this analytical time evolution solution and presents two derivation methods. Then the paper further generalizes to obtain more non-series analytical solutions for time evolution oscillator, and finds the recursive relation to obtain these analytical solutions. Finally, the educational significance and applications of these analytical solutions are discussed.

DOI
10.26599/phys.2024.9320416

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In 1926, E.Schrödinger found an analytical solution for the time evolution of the one-dimensional harmonic oscillator Schrödinger Equation. This solution is not in the form of a summation of energy eigenstate wave functions, which provides a new perspective on the time evolution in quantum mechanics. This paper mainly analyzes this analytical time evolution solution and presents two derivation methods. Then the paper further generalizes to obtain more non-series analytical solutions for time evolution oscillator, and finds the recursive relation to obtain these analytical solutions. Finally, the educational significance and applications of these analytical solutions are discussed.
A method for calculating the analytical solutions of the one-dimensional Schr\"odinger equation is suggested. A general discussion of the possible forms of the potentials and wave functions that are necessary to get the analytical solution is presented. In general, the analytical solutions appear in multiplets corresponding to the quantum number n of the harmonic oscillator. As an application, known solutions for the anharmonic oscillators are critically recalculated and a few additional results are found. Analytical solutions are also found for the generalized Morse oscillators. \textcopyright{} 1996 The American Physical Society.
The time evolution of a quantum harmonic oscillator with a series of sudden jumps of the mass or the frequency is determined in the form of a recursion relation. An approximating method is developed for determining the time evolution of harmonic oscillators with arbitrary derivable functions of the frequency or the mass. The approximate solution is shown to tend to the analytical one in the limiting case. As a demonstration of the approximating method, the solution of the problem of damped oscillation in the square of the oscillator frequency is presented.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.
A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptomatic solution of the Schrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained. This equation has a recursive property which has been exploited to obtain the solutions of the equation. It is all about reconstruction of the equation with different solutions of that unknown function by repetitive differentiation and that ultimately leads to the wave functions of the Schrodinger equation for harmonic oscillator.