The math generalizes!īut harmonic oscillation has value far beyond the case of a mass on a spring. The simplest example of this is a mass on a spring, so we will treat that example in great and gory detail. It's called "harmonic" because the solution of Newton's second law (a second order differential equation that determines the motion of the object) are sines and cosines of time with a particular frequency - just like the result produced by a pure musical tone heard at a particular point in space.
![simple harmonic oscillator simple harmonic oscillator](https://img.youtube.com/vi/ZcZQsj6YAgU/0.jpg)
(A quantum system never can sit exactly at rest at a minimum point, which is why we draw our molecular bound states not exactly at the bottom of the potential well.) If the restoring force can be treated as linear - or equivalently, if the potential energy can be treated as a parabola - then the motion is called harmonic. The (classical) system will sit at the minimum point when it is stable. If there is a stable point, then the system's potential energy should look like a well with a minimum point. We'll go through this in detail in specific cases so it might make more sense then. The force will then build up to slow it down, but not before it has gone the other half of the oscillation.
![simple harmonic oscillator simple harmonic oscillator](https://quantummechanics.ucsd.edu/ph130a/130_notes/img1270.png)
There are three core situations that lead to a system oscillating. The powerful tools of spectral and Fourier analysis, used in many biological applications, are based on the math of the simple harmonic oscillator like the mass on a spring. Perhaps even more important, the math of a single oscillator is extended to infinitely many oscillators to allow us to develop the mathematics of waves, including sound, light, and the propagation of nerve signals. Our modest little model is the toy model that provides insight into the mathematics that describe all oscillations. Oscillations are even observed in a variety of chemical reactions. Coupled oscillators (especially non-linear ones) are the basis of important models of the functioning of neural nets in the brain. They occur in biological organisms in the detection of sound in the ear, the beating of the heart, and they are the critical phenomenon in the functioning of the brain. The reason we do this problem in such great detail - considering multiple similar systems, adding damping, adding driving forces, doing all the math - is because, while the mass on a spring system is rarely seen, oscillations are everywhere.
![simple harmonic oscillator simple harmonic oscillator](https://www.rfwireless-world.com/images/Electrical-Oscillator.jpg)
We almost never encounter a mass hanging on a spring in everyday life and certainly not in biology. The modest little problem of a mass hanging on a spring is covered in almost every introductory physics class.