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Journal of Chemical Education

A Heuristic Derivation of the Linear Harmonic Oscillator Energy Levels

Michael G. Messina

Published Apr 1, 2000• 0 citations
Publisher

Abstract

In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrödinger equation for the harmonic oscillator. This derivation treats the harmonic oscillator as a quasiparticle in a box and yields qualitative results, these being the correct scaling of the energy with quantum number and vibrational frequency, but does not yield the exact result. By comparing the exact result with the result of the heuristic derivation students are introduced to the idea of tunneling in quantum mechanics and how tunneling can effect the energy levels of quantum mechanical systems.

DOI
10.1021/ed077p492

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In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrödinger equation for the harmonic oscillator. This derivation treats the harmonic oscillator as a quasiparticle in a box and yields qualitative results, these being the correct scaling of the energy with quantum number and vibrational frequency, but does not yield the exact result. By comparing the exact result with the result of the heuristic derivation students are introduced to the idea of tunneling in quantum mechanics and how tunneling can effect the energy levels of quantum mechanical systems.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
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.
Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.
Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.
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.
Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.