You may be familiar with several examples of harmonic oscillators form classical mechanics, such as particles on a spring or the pendulum for small deviation from equilibrium, etc.
Let me look at the characteristics of one such example, a particle of mass on a spring. When the particle moves a distance away from the equilibrium position , there will be a restoring force pushing the particle back ( right of equilibrium, and on the left). This can be derived from a potential
(7.1) |
Actually we shall write . The equation of motion
(7.2) |
has the solution
(7.3) |
We now consider how this system behaves quantum-mechanically.