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## Harmonic Oscillators

Today I was planning to post the fourth and final part of my series of blogs about the derivation of Planck’s radiation law. But, I realised on Sunday that it would not be ready, so I’m postponing it until next Thursday (17th). Parts 1, 2 and 3 are here, here and here respectively).

One of the reasons for this is that my time is being consumed by writing articles for 30-second Einstein which I talked about on Tuesday. Another reason is that I am scrambling to finish a slew of things by the 15th (next Tuesday!!), as I have been asked to go on another cruise to give astronomy lectures. More about that next week ðŸ™‚

There is a third reason, I have realised that I have not yet done a blog about harmonic oscillators, which is a necessary part of understanding Planck’s derivation. So, that is the subject of today’s blogpost.

Harmonic oscillators is another term for something which is exhibiting simple harmonic motion, and I did blog here about how a pendulum exhibits simple harmonic motion (SHM), and how this relates to circular motion. Another example of SHM is a spring oscillating back and forth. Whether the spring is vertical or horizontal, if it is displaced from its equilibrium position it will exhibit SHM. So, a spring is a harmonic oscillator.

## The frequency of a harmonic oscillator

The restoring force on a spring when it is displaced from its equilibrium position is given by Hooke’s law, which states

$\vec{F} = - k \vec{x}$

where $\vec{F} \text{ and } \vec{x}$ are the force and displacement respectively (vector quantities), and the minus sign is telling us that the force acts in the opposite direction to the displacement; that is it is a restoring force which is directed back towards the equilibrium position. The term $k$ is known as Hooke’s constant, and is basically the stiffness of the spring.

Because we can also write the force in terms of mass and acceleration (Newton’s 2nd law of motion), and acceleration is the second derivate of displacement, we can write

$m \vec{a} = m \frac{ d^{2}\vec{x} }{ dt^{2} }= - k \vec{x}$

If we divide by $m$ we get an expression for the acceleration, which is

$\boxed{ \vec{a} = - \frac{k}{m} \vec{x} }$

which, if you compare it to the equation for SHM for a pendulum, has the same form. The usual way to write equations of SMH is to write

$\vec{a} = - \omega^{2} \vec{x}$

where $\omega$ is the angular velocity, and as I mentioned in the blog I did on the pendulum, $\omega$ is related to the period of the SHM, via $T = 2 \pi / \omega$.

For our derivation of Planck’s radiation law, the parts which we need to know about are that the frequency of a harmonic oscillator, $\nu$ is given by $\nu = \omega / 2 \pi$, and so depends only on $\omega, \; \boxed{\nu \propto \omega }$. So the frequency of the spring’s oscillations depends only on k/m, we can write $\boxed{ \nu \propto k/m }$. A stiffer spring oscillates with a higher frequency, more mass (in either the spring or what is attached to it) will reduce the frequency of the oscillations.

## The energy of a harmonic oscillator

The other thing we need to know about to understand Planck’s derivation of his blackbody radiation law is the energy of the harmonic oscillator. This is always constant, but is divided between kinetic energy and potential energy. The kinetic energy is at a maximum when the spring is at its equilibrium position, at this moment it actually has zero potential energy.

The velocity of a harmonic oscillator $v$ can be found my differentiating the displacement $x$ with respect to time. The expression for displacement (see my blog here on SHM in a pendulum) is

$x(t) = A sin ( \omega t )$

where $A$ is the maximum displacement (amplitude) of the oscillstions. So

$v = \frac{dx}{dt} = A \omega cos ( \omega t)$

This will be a maximum when $cos( \omega t) = 1$ and so

$v_{max} = A \omega$

which means that the maximum kinetic energy, and hence the total energy of the harmonic oscillator is given by

$\text{Total energy} = E = \frac{1}{2}mv_{max}^{2} = \frac{1}{2} m A^{2} \omega^{2}$

As the frequency $\nu$ is just $\omega / 2 \pi$, this tells us that, for a harmonic oscillator of a given mass, the energy depends on both the square of frequency $\nu$ and the square of the size of the oscillations (larger oscillations mean more energy, double the size of the oscillations and the energy goes up by a factor of four). Mathematically we can write this as $\boxed{ E \propto A^{2} }$ and $\boxed{ E \propto \nu^{2} }$.

As we shall see, the theoretical explanation which Planck concocted to explain his blackbody curve involved assuming the walls of the cavity producing the radiation oscillated in resonance with the radiation; this is why I needed to derive these things on this blog today.