The Fourier transform has countless applications across science and math, but its role in quantum mechanics is particularly profound. It connects the position space wavefunction and the momentum space wavefunction, offering deep insights into the nature of quantum particles. In this lesson, Dr. Jacob Hudis provides detailed examples, starting with the Fourier series and then extending to the Fourier transform. He explains how complex exponentials form a basis set that can construct any function, making the concept more intuitive than the traditional sum of sines and cosines. Dr. Hudis makes comparisons between discrete vectors and continuous functions to clarify these challenging concepts. The goal is to provide a clear and accessible understanding.