Why study the SHO?

Figure 1.5:

What happens to a system when it is disturbed from stable equilibrium? This question that arises in a large variety of situations. For example, the atoms in many solids (eg. NACl, diamond and steel) are arranged in a periodic crystal as shown in Figure 1.5. The periodic crystal is known to be an equilibrium configuration of the atoms. The atoms are continuously disturbed from their equilibrium positions (shown in Figure 1.5) as a consequence of random thermal motions and external forces which may happen to act on the solid. The study of oscillations in the atoms disturbed from their equilibrium position is very interesting. In fact the oscillations of the different atoms are coupled, and this gives rise to collective vibrations of the whole crystal which can explain properties like the specific heat capacity of the solid. We shall come back to this later, right now the crucial point is that each atom behaves like a SHO if we assume that all the other atoms remain fixed. This is generic to all systems which are slightly disturbed from stable equilibrium.

We now show that any potential $V(x)$ is well represented by a SHO potential in the neighbourhood of points of stable equilibrium. The origin of $x$ is chosen so that the point of stable equilibrium is located at $x=0$. For small values of $x$ it is possible to approximate the function $V(x)$ using a Taylor series

where the higher powers of $x$ are assumed to be negligibly small. We know that at the points of stable equilibrium the force vanishes ie. $F=-d V(x)/dx=0$ and $V(x)$ has a minima

$$k=\left(\frac{d^2 V(x)}{dx^2} \right)_{x=0}>0 \,.$$ (1.17)

This tells us that the potential is approximately

$$V(x) \approx V(x)_{x=0} + \frac{1}{2} k x^2$$ (1.18)

which is a SHO potential. Figure 1.6 shows two different potentials which are well approximated by the same SHO potential in the neighbourhood of the point of stable equilibrium. The oscillation frequency is exactly the same for particles slightly disturbed from equilibrium in these three different potentials.

Figure 1.6:

The study of SHO is important because it occurs in a large variety of situations where the system is slightly disturbed from equilibrium. We discuss a few simple situations.