# Analytical Method for Solving the Simple Harmonic Oscillator

### From FSUPhysicsWiki

In addition to the elegant operator-based method described previously to solve the harmonic oscillator, one may also write down and solve the Schrödinger equation for the system. The equation is

If we now introduce the dimensionless position, and the dimensionless energy, then this reduces to the much simpler form,

Let us now look at the asymptotic behavior of the wave function. For large values of our equation may be approximated as

We now make the following ansatz for its solution:

Substituting this in the asymptotic equation, we get

or, in the limit of large

with this value of k,

For ψ(ξ) to remain finite at the origin, we must set Therefore, the long-distance behavior of the wave function is given by

Now that we have separated out the asymptotic behavior, we now postulate that the complete solution, valid everywhere, can be written as

where *h*(ξ) is a function that diverges more slowly than for large ξ. The first and second derivatives of this form are

and

Substituting these into the differential equation, we get

Let us try a series solution for

Substituting this into the differential equation and collecting like powers yields

Equating the coefficients of each power of on both sides, we obtain the recursion relation,

For large values of this may be approximated as

If we assume that is only finite for even then this is satisfied by

This yields the power series for which diverges more rapidly for large than Taking only odd coefficients yields a similar result, namely the power series for

where erf(*x*) is the error function. This means that, to obtain a wave function that goes to zero as we must truncate the power series at a finite order; i.e., must be a polynomial. To truncate the series to order where is a non-negative integer, we must set

and take only those coefficients for which has the same "parity" (odd or even) as The fact that the resulting polynomial will either be an even or odd function could have been seen from the fact that the system under investigation has an even potential. The above condition yields the allowed energy levels for the system,

With this constraint on ε, the recursion relation for the power series coefficients becomes

This recursion relation will yield the Hermite polynomial,

The properly normalized eigenfunctions are

as we found in the previous section.

## Problem

(Schaum, *Theory and Problems of Quantum Mechanics*, Chapter 5)

Consider a particle with charge in a three-dimensional isotropic harmonic potential,

and in the presence of a constant electric field, Find the energy levels and eigenstates of the patricle.

## Reference

Introduction to Quantum Mechanics, 2nd ed. , by D. J. Griffiths