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.

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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

$$V\left(x\right)=\frac{1}{2}k{x}^2.$$

(7.1)

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

$$mẍ=-m{\omega }^{2}x$$

(7.2)

has the solution

$$x\left(t\right)=Acos\left(\omega t\right)+Bsin\left(\omega t\right).$$

(7.3)

We now consider how this system behaves quantum-mechanically.

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