Math as Music
Joseph Fourier discovered that any periodic function can be written as a sum of simple sines and cosines. This is like saying any complex sound is just a combination of pure musical notes.
\[f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]\]
Worked Examples
Example 1: The Square Wave
A square wave (on/off) can be built by adding up odd harmonics: \(\sin(x) + \frac{1}{3}\sin(3x) + \frac{1}{5}\sin(5x) + \dots\). The more terms you add, the sharper the corners of the square become.
The Bridge to Quantum Mechanics
In Quantum Mechanics, this decomposition is called Superposition. Any possible state of a quantum system can be written as a sum of its energy eigenstates. Just as the Fourier series finds the "notes" in a sound, the Schrödinger Equation finds the "energy notes" in an atom. This is why we can describe a particle's state as a "Vector" in a space of many possible waves.