Chapter 7
The Harmonic oscillator

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.


Figure 7.1: The mass on the spring and its equilibrium position

Let me look at the characteristics of one such example, a particle of mass m on a spring. When the particle moves a distance x away from the equilibrium position x 0, there will be a restoring force k x pushing the particle back ( x> 0 right of equilibrium, and x< 0 on the left). This can be derived from a potential

V ( x ) = 1 2 k x 2 . (7.1)

Actually we shall write k= mω 2. The equation of motion

m = m ω 2 x (7.2)

has the solution

x ( t ) = A cos ( ω t ) + B sin ( ω t ) . (7.3)

We now consider how this system behaves quantum-mechanically.