Non-Periodic Functions
If a function doesn't repeat (like a single pulse), we use the Fourier Transform. Instead of a sum of discrete notes, we have an integral over a continuous spectrum of frequencies.
\[\mathcal{F}\{f(x)\} = F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx\]
This transforms "Space" (\(x\)) into "Wavenumber" (\(k\)).
Worked Examples
Example 1: The Delta Pulse
The Fourier transform of a single sharp spike (Dirac Delta) is a flat horizontal line. This means a perfectly localized spike contains every possible frequency in equal amounts.
The Bridge to Quantum Mechanics
The Fourier transform is the heart of Wave-Particle Duality. The wavefunction in position space \(\psi(x)\) and the wavefunction in momentum space \(\phi(p)\) are Fourier transforms of each other. Since \(p = \hbar k\), the transform literally converts "Where it is" into "How fast it's moving." This is why a particle with a precise position must have a completely spread-out (uncertain) momentum—the math of the Fourier transform doesn't allow both to be spikes at the same time.