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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
\includegraphics[width=3cm]{Figures/spring.eps}

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 x0, 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) = $ {\frac{1}{2}}$kx2. (7.1)

Actually we shall write k = m$ \omega^{2}_{}$. The equation of motion

m$\displaystyle \ddot{x}$ = - m$\displaystyle \omega^{2}_{}$x (7.2)

has the solution

x(t) = Acos($\displaystyle \omega$t) + Bsin($\displaystyle \omega$t). (7.3)

We now consider how this system behaves quantum-mechanically.



 
next up [*]
Next: 7.1 Dimensionless coordinates Up: Quantum Mechanics I Previous: 6.3 Square barrier

© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk