Figure 7.1: The mass on the spring and its equilibrium position
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 m on a spring. When the particle moves a distance x away from the equilibrium position {x}_{0}, there will be a restoring force − kx 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\over
2}k{x}^{2}.
| (7.1) |
Actually we shall write k = m{ω}^{2}. The equation of motion
|
m\ddot{x} = −m{ω}^{2}x
| (7.2) |
has the solution
|
x(t) = A\mathop{cos}\nolimits (ωt) + B\mathop{sin}\nolimits (ωt).
| (7.3) |
We now consider how this system behaves quantum-mechanically.