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Journal of Chemical Education

Intuitive Solution to Quantum Harmonic Oscillator at Infinity

Cory Christopher Pye

Published Jun 1, 2004• 1 citation
Publisher

Abstract

In the solution to the quantum harmonic oscillator problem, the approximate Gaussian solution at large displacements is typically presented without proof. While it is relatively easy to verify that this form is an approximate solution, it is not obvious how this form can be obtained. What follows is a simple intuitive argument that gives the exact Gaussian form required.

DOI
10.1021/ed081p830

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In the solution to the quantum harmonic oscillator problem, the approximate Gaussian solution at large displacements is typically presented without proof. While it is relatively easy to verify that this form is an approximate solution, it is not obvious how this form can be obtained. What follows is a simple intuitive argument that gives the exact Gaussian form required.
Abstract This paper advocates writing the eigenfunctions of the harmonic oscillator without Hermite polynomials, by means of the following recurrence relation: Ψ n = 2 α / n x Ψ n − 1 − ( n − 1 ) / n Ψ n − 2 ( n > 0 ) , with Ψ 0 = ( α / π ) 1 / 4 e − α x 2 / 2 . It is easy to prove that these functions are eigenfunctions of the harmonic oscillator. It is shown that many of the exercises related to the harmonic oscillator found in textbooks can be solved in a simple and elegant way by using this recurrence relation.
It is a rather curious fact that the exact quantal treatment of forced oscillation has only recently been discovered to be feasible. There may be a lot of general arguments which make the possibility of such a treatment very convincing, but I myself find the ground for it in an elementary observation that the wave equation of forced harmonic oscillation, including the case of subharmonic resonance, admits a solution of Gaussian type, i.e. of the form: an exponential of a quadratic. On this basis the complete solution and the related problems are analyzed here.
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.
The purpose of this work is to introduce, in a simple, intuitive way, the coherent and squeezed states of the quantum harmonic oscillator (QHO), through a series of exercises, which are solved in detail. Starting from the application of a spatial translation to the ground state of the QHO, we introduce the spatial and momentum translations, focusing on their application to the QHO, which leads us to the displacement operator. Next, we introduce the coherent states and examine their basic aspects. We then proceed to give a simple and purely intuitive introduction to the squeezed states and we conclude by identifying the coherent states as states of minimum energy expectation value compared to the respective squeezed states. The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the QHO.
By solving exactly with a simple method the time dependent Schrödinger equation for simple harmonic oscillator, we demonstrate that the probability density associated with any displaced eigenstate has the following properties: (a) Its centroid oscillates according to the classical law. (b) Its wave packet form remains rigid. The same method is also applied to solve exactly the problem of simple harmonic oscillator with the force constant depending on time. In this case, probability densities with the same wave packet structures as that of the usual simple harmonic oscillator can be found to move in such a way that their centroids follow the classical law but their wave packets deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.