Classic Harmonic Oscillator

Let's say something is an oscillator.

We have this restoring force

E=p22m+12mω2x2E = \frac{p^2}{2m} + \tfrac{1}{2}m\omega^2 x^2

where

k≜mω2k\triangleq m\omega^2

This is similar to a oscillator.

F=−kxF = -kx

Any smooth potential with a minimum at x0x_0 expands as (via Taylor Series)

V(x)=V(x0)+(x−x0)V′(x0)+12(x−x0)2V′′(x0)+⋯V(x) = V(x_0) + (x-x_0)V'(x_0) + \tfrac{1}{2}(x-x_0)^2 V''(x_0) + \cdots =constant+12k(x−x0)2= \text{constant} + \tfrac{1}{2}k(x - x_0)^2

where

k≜V′′(x0)k\triangleq V''(x_0)

Applications

  1. vibrations in materials and molecules
  2. trapped atoms and ions (quantum information processing)
  3. light/photons
  4. circuit oscillator (LC)