To Understand the Fourier Transform, Start From Quantum Mechanics

Develop a deep understanding of the Fourier transform by appreciating the critical role it plays in quantum mechanics! Get the notes for free here: Sign up for my newsletter for additional physics lessons: The Fourier transform has a million applications across all sorts of fields in science and math. But one of the very deepest arises in quantum mechanics, where it provides a map between two parallel descriptions of a quantum particle: one in terms of the position space wavefunction, and a dual description in terms of the momentum space wavefunction. Understanding this connection is also one of the best ways of learning what the Fourier transform really means. We’ll start by thinking about the quantum mechanics of a particle on a circle, which requires that the wavefunction be periodic. That lets us expand it in a Fourier series---a superposition of many sine and cosine functions, or eq