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arXiv:1705.10293 · American Journal of Physics

Solutions of the Schroedinger equation for piecewise harmonic potentials: Remarks on the asymptotic behavior of the wave functions

Francisco Diego Mazzitelli, Martín D. Mazzitelli, Pedro I. Soubelet

Published Sep 18, 2017• 3 citations
arXivHTMLPDF

Abstract

We discuss the solutions of the Schroedinger equation for piecewise potentials, given by the harmonic oscillator potential for |x|>a and by an arbitrary function for |x|<a, using elementary methods. The study of this problem sheds light on usual errors made in discussions of the asymptotic behavior of the eigenfunctions of the quantum harmonic oscillator and can also be used for the analysis of the eigenfunctions of the hydrogen atom. We present explicit results for the energy levels of a potential of this class, used to model the confinement of electrons in nanostructures.

DOI
10.1119/1.4994808

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We discuss the solutions of the Schroedinger equation for piecewise potentials, given by the harmonic oscillator potential for |x|>a and by an arbitrary function for |x|<a, using elementary methods. The study of this problem sheds light on usual errors made in discussions of the asymptotic behavior of the eigenfunctions of the quantum harmonic oscillator and can also be used for the analysis of the eigenfunctions of the hydrogen atom. We present explicit results for the energy levels of a potential of this class, used to model the confinement of electrons in nanostructures.
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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 purpose of this work is to show the stability of the hydrogen atom with the use the Quantum Oscillatory Modulated Potential and the Heisenberg equations of motion, postulating that the electron in the hydrogen atom is behaving as a quantum harmonic oscillator. With the electron confined between two potential barriers, created by the new potential function, we are considering that at absolute temperature the power absorbed or emitted by the electron per unit of time can be used to determine the zero point energy of the oscillator. Assuming that electron is only exchanging energy with the nucleus of the atom we are making use of the operators of creation and annihilation of a photon to explain how the energy between the proton and the electron can be exchanged to keep the atom a stable system.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.