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17.1: Particle in a Well

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We begin with the one-dimensional case of a particle oscillating about a local minimum of the potential energy V(x). We’ll assume that near the minimum, call it x0 the potential is well described by the leading second-order term, V(x)=12V(x0)(xx0)2 so we’re taking the zero of potential at x0, assuming that the second derivative V(x0)0, and (for now) neglecting higher order terms.

Particle in a curvy double well. Particle is at the minimum of the upper well.
Figure 17.1.1

To simplify the equations, we’ll also move the x origin to x0, so

m¨x=V(0)x=kx

replacing the second derivative with the standard “spring constant” expression.

This equation has solution

x=Acos(ωt+δ), or x=Re(Beiωt),B=Aeiδ,ω=k/m

(This can, of course, also be derived from the Lagrangian, easily shown to be L=12m˙x212mω2x2.


This page titled 17.1: Particle in a Well is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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