Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

9.1: Harmonic oscillators

( \newcommand{\kernel}{\mathrm{null}\,}\)

One of the major playing fields for operatorial methods is the harmonic oscillator. Even though they look very artificial, harmonic potentials play an extremely important role in many areas of physics. This is due to the fact that around an equilibrium point, where the forces vanish, any potential behaves as an harmonic one (plus small corrections). This can best be seen by making a Taylor series expansion about such a point,

V(x)=V0+12mω2x2+O(x3).

Question

Why is there no linear term in Equation ????

For small enough x the quadratic term dominates, and we can ignore other terms. Such situations occur in many physical problems, and make the harmonic oscillator such an important problem.

As explained in our first discussion of harmonic oscillators, we scale to dimensionless variables (“pure numbers”)

y=mωx

with ϵ=Eω.

In these new variables the Schrödinger equation becomes

12(d2dy2+y2)u(y)=ϵu(y).


This page titled 9.1: Harmonic oscillators is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?