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Journal of Chemical Education

Exploring the Harmonic Oscillator Wave Functions

Theresa Julia Zielinski

Published Jul 1, 2007• 0 citations
Publisher

Abstract

The goal of this document is to have students explore the solutions to the quantum mechanical harmonic oscillator Schrödinger equation by examining the exponential component, generating the various Hermite polynomials, and then creating the final HO wave functions as a product of a normalization factor, the exponential component, and a Hermite polynomial.

DOI
10.1021/ed084p1232.1

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The goal of this document is to have students explore the solutions to the quantum mechanical harmonic oscillator Schrödinger equation by examining the exponential component, generating the various Hermite polynomials, and then creating the final HO wave functions as a product of a normalization factor, the exponential component, and a Hermite polynomial.
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.
This chapter exclusively deals with the application of Schrödinger equation on various systems such as the particle in a box; the particle in a rectangular three-dimensional box; energy levels for a cubic potential box; the tunnel effect; the final equation for tunnel effect, that is, transparency factor which gives a clue to understand that the probability of transmission decreases rapidly as the width of the barrier increases; quantum mechanical explanation of emission of α -particle; the particle on a ring considering spherical polar co-ordinates; and the particle on sphere. This chapter also comprises the Legendre polynomials, associated Legendre equation, associated Legendre function and spherical harmonics, Hermite polynomials and its applications, and orthogonal properties of Hermite polynomials. It also provides the standard topics like simple linear harmonic oscillator, and its classical and quantum mechanical treatment which also includes the asymptotic and series solutions and energy levels of a linear harmonic oscillator. The Schrödinger equation also plays with rigid rotors comprising its rigorous treatment and also the F and T equations which are very important. The chapter ends with the references, standard several computational problems, questions on concepts, and also standard numerical problems for the benefits of the readers.
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 asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.