Fourier analysis applied to quantum mechanics and the uncertainty principle
Quantum Mechanics (Winter, 2012)
Lecture Playlist

Introduction to quantum mechanics

The basic logic of quantum mechanics

Vector spaces and operators

Time evolution of a quantum system

Uncertainty, unitary evolution, and the Schrödinger equation

Entanglement

Entanglement and the nature of reality

Particles moving in one dimension and their operators

Fourier analysis applied to quantum mechanics and the uncertainty principle

The uncertainty principle and classical analogs
Professor Susskind opens the lecture with a review of the entangled singlet and triplet states and how they decay. He then shows how Fourier analysis can be used to decompose a typical quantum mechanical wave function.
He then continues the discussion of a continuous system  a single particle moving in one dimension  and shows that the solutions to the eigenvector equations for position and momentum lead to the uncertainty principle. In other words, the wave function solution for a specific value of momentum has probabilities for the position everywhere (in the single dimension). This derivation shows that the position and momentum wave functions are Fourier transforms of each other. Thus mathematically the uncertainty principle is simply a a statement about Fourier transforms.
 Triplet state decay
 Fourier analysis applied to quantum mechanics
 Relationship between the Fourier transform and the uncertainty principle